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Chapter 2 Topological Vector Spaces
CHAPTER 2 TOPOLOGICAL VECTOR SPACES
DEFINITION 2 . 1 :
( F , T ~ b) e a t o p o l o g i c a l d i v i s i o n r i n g . By a
Let
f o p o t u g i c a l ? u e c f o h hpUCe ( T V where on
E
S)
i s a vector s p a c e o v e r
o u e h ( F , f F ) w e mean a p a i r ( E , T ) , F, and T i s a T V S Zopok?ogy
t h a t is
E;
(a)
t h e mapping ( x , y ) + x + y
from ( E J )
( E , T ) to (E,T)
x
i s continuous ;
t h e mapping (X,x) +Ax
(b)
from ( F , T ~ ) ( E , T ) t o ( E , T ) i s
continuous. If
is a
i s d e f i n e d by a n a b s o l u t e v a l u e
T~
1).
(F,1 .
X
IA 1 ,
+
(F,-cF), w e s h a l l say t h a t ( E J )
TVS over
is a
and ( E l T) TVS ~ U W L
C l e a r l y , any t o p o l o g i c a l d i v i s i o n r i n g i s a TVS o v e r
it-
self. If
( E , T ) i s a T V S o v e r (F,rF) a n d
T
# {@,E), w e say t h a t
(E,T) i s a phvpeh T VS o v e r ( F , T ~ ) i; f T = {@,E}, the TVS (EJ) i s c a l l e d i m p h o p e h . Notice t h a t (E,T) i s a n i m p r o p e r T V S i f , and o n l y i f ,
0
C l e a r l y , any non-zero Hausdorff T V S
= E.
(E,T) is a proper TVS.
L e t ( E , T ) b e a T V S o u c h ( F , T ~ )Foh . each xoE E F*, ,the m a p p i n g 3 x x + x and x A0x ate 0
PROPOSITION 2.2: and e a c h
ho
E
homeomvhphihmn a 4
-+
( E , T ) o n X o i.tnek?d.
x
PROOF: The mapping If
ho # 0 , t h e n
too, again
+
x
+
+
Aox
hilx
by D e f i n i t i o n
i s c o n t i n u o u s from is
2.1.
Definition2.1.
i t s i n v e r s e , which i s c o n t i n u o u s In particular,
x
-+
- x
is a
PROLLA
32
homeomorphism. x x + x0 The c o n t i n u i t y of Its i n v e r s e i s t h e mapping x -+ x
follows f r o m D e f i n i t i o n 2 . 1 . = x + (- xo).
-+
- x0
L e t (E,T) b e a T V S o v c 4 (F,rF) and a E E. T h e n 0 i d , and onLy i d , a + V i b u n e i g h b a ~ h o u d ad a. Any b u b b e t V C E i n a n e i g h b o h h o o d 0 6 0 id, and o n L g id, - V i n a neighbntrhood ad 0. Mohe genetraLLy,id X t F*, V C E i b a neighbohhood a d 0 , id und o n L y id, AV i n a ncighbohhaod a6 0 . COROLLARY 2.3:
v c E i n a neighboahood o h
DEFINITION 2.4: A s u b s e t c a l l e d nymme,thic i f
S =
of a vector s p a c e E o v e r
S C E
-
F is
S.
T h e hymmekhic n e i g h b u h h u o d n (E,r) dohm a b a b i b 04 neighbahhuodn a i 0 .
i n a
a6 0
PROPOSITION 2.5:
PROOF: L e t V b e a n e i g h b o r h o o d of 0 i n (E,T). 2 . 3 , - V i s a n e i g h b o r h o o d o f 0 . Hence V n r i c n e i g h b o r h o o d of 0 , w h i c h i s c o n t a i n e d i n
(-
By
T VS
Corollary
V) i s a symmetV.
DEFINITION 2.6: L e t (F,I*I)b e a v a l u e d d i v i s i o n r i n g . A s u b s e t of a v e c t o r s p a c e
S C E
each
x
6 > 0
E l there is a
E
implies
x
E
E over
F is called
a b b o h b i n g i f , for
s u c h t h a t f o r any
h
E
F, ( X ( z 6
AS.
C l e a r l y , i f S i s a b s o r b i n g , and
T 3 S,
then T i s absorbing
too.
PROPOSITION 2.7:
ncighbotrhuod u d
Let ( E , T ) b e a T V S o v e ~(F, 1 0 i n (E,r) i n abborrbing.
V C E
PROOF: L e t
b e a n e i g h b o r h o o d of
b e g i v e n . The mapping
X
t h e r e is
E
6
0 , Then ( A / > 6
= E - ~>
> 0
+
such t h a t
DEFINITION 2.8: L e t ( F ,
1- 1)
Ax Ihl
0
- 1).
Then
evetry
.
i n (E,r) L e t x 6 E
i s c o n t i n u o u s a t t h e o r i g i n . Hence 5 E implies Ax E V. Let
implies
X-lx
E V,
i.e.
x
E
XV.
be a v a l u e d d i v i s i o n r i n g . A s u b s e t
33
TOPOLOGICAL VECTOR SPACES
S
of
E
C
a vector space E over
1x1
for e v e r y Clearly,
S i s b a l a n c e d a n d non-empty,
if
Let
(F, / *
1).
{xu} i n
-
S
S.
then
The cLvnuke o d a bulanced A E X
x
s,
E
and
Axc,
Now
a balanced s u b s e t of a
be
S C E
Let
baLanced i f
AS C
S
< I.
PROPOSITION 2 . 9 : PROOF:
F is called
S
E
1x1
<
and
i h
0 E S.
balanced.
T VS
(EJ)
over
1. Then x i s t h e l i m i t o f a n e t Axc, Ax. T h e r e f o r e Ax E 5, a n d +
i s balanced.
If
{Ax; j h l 5 1, x
then
S C El
balanced set containing
E S }
is t h e
smallest
S. I t i s c a l l e d t h e baLanced h u l l o d S .
C l e a r l y , t h e u n i o n of a n y f a m i l y of b a l a n c e d s e t s a n c e d . Hence, t h e r e e x i s t s for a n y s e t
if
bal-
S C E l a b i g g e s t balanced
s e t K c o n t a i n e d i n S. K i s c a l l e d t h e b a l a n c e d hekneR i t i s non-empty i f , a n d o n l y i f , 0 E S.
06
S,
L e t ( F , I 1 ) b~ a non-ttiviaLLy vaBued d c v h L o n king, and l e t ( E , T ) be a T V S owe& (F,1 . 1 ) . T h e baLanced hefine& 06 aviy tieighbokhood o h 0 i n a n e i g h b o t h o v d 0 6 0.
PROPOSITION 2 . 1 0 :
PROOF: L e t V b e a n e i g h b o r h o o d of 0 i n ( E , T ) . S i n c e ( A , x )
i s c o n t i n u o u s a t (0,O) of
0
(EJ)
Choose of
0,
in
V.
AU =
and
0 < and
such t h a t
1 Xo 1 5
XoW
C
,
there exist
1x1 5
6
+
6 > 0 and neighborhood
x
and
E W
Ax
imply
Xx W
E V.
i s a neighborhood V . Now t h e b a l a n c e d h u l l of AoW i s c o n t a i n e d
Indeed, i f
6 . By C o r o l l a r y 2 . 3 ,
u E AoW, t h e n
u
=
AoW
X 0w ,
with
w
E W.
Hence
hhow E V f o r a n y l h l 5 1, b e c a u s e lhhol = 1x1 * /Xol 5 6 w E W. T h i s shows t h a t A 0 W C K C V , w h e r e K i s t h e b a l -
a n c e d k e r n e l of
V.
COROLLARY 2.11: L e t ( E , r ) b e a T V S awe4 a n u n - , t h i v i a & ? y v a l u e d
d i v i n i v n hing (F, 1 . I ) . T h e balanced neighbohhaadh b a n i d 0 6 neighbohhoodn a t 0.
PROPOSITION 2 . 1 2 :
L e t ( E , T ) be a T V S O v e h
UQ
0 dotm a
( F , T ~ ) T. h e
cloned
34
PROLLA
neighbuhkoodn 0 4
a b a n i o oA ne.i.ghba,thoodn a t
0
L e t V be a n e i g h b o r h o o d of 0 i n ( E l T )
PROOF:
is continuous a t ( O , O ) ,
such t h a t hood of
x
0,
v
Choose v = x
+
W
- W)
(x
E
- w
f o r some
W e h a v e shown t h a t
w
neighborhood
x
k.
E
Since
is a n e i g h b o r h o o d of
W
w E W.
of
W
C
v
Then
n W.
w
.
S i n c e (x,y)
+
t W;
W C V.
Hence
=
v
v
+
0
Hence (x - W) n W # @.
and
x
x+y
is a neighbor-
-W
x.
Hence
+
W of
t h e r e is another neighborhood
L e t now
W C V.
-
0.
do~111
E
x - W implies t h a t
w, t h a t i s
V
x E W + W.
contains the closed
0.
l e t ( E , ? ) b c a T V S t ~ ~ ea hn v n - t h i v i a ! - L y v a e u e d I ) . T h e h e e x i n d n a b u n i n a d neighboahaad6 ad 0 c o n n i a t i n g 06 c l o n e d and b a l a n c e d n e t n , namely x h e s e t 0 4 a U cloned and balanced ncighbothoudn 0 6 0 .
COROLLARY 2 . 1 3 : d i v i n i u n Ring
PROOF: L e t
(F, 1 .
0 i n (E,r). By
V b e a n e i g h b o r h o o d of
t h e r e e x i s t s a closed neighborhood
2.12, V.
By C o r o l l a r y 2 . 1 1 ,
0
contained i n
W.
contained
0
in
t h e r e e x i s t s a balanced neighborhood U of
Hence
0 By P r o p o s i t i o n 2 . 9 ,
W of
Proposition
U
E U C T S C
wc
v.
is balanced; i.e.
b a l a n c e d n e i g h b o r h o o d of
0 contained i n
U
is
a
closed
and
V.
L e t S C E b e a b a l a n c e d o u b s e t 06 a vccxUh n p a c e E a v e h a n a n - t h i u i a l l y valued d i v i n i o n h i n g (F,1 . 1 ) . Then S i n ubnvn.Ling i . 4 , and o n l y id, g i v e n x E E thetie exist5 E F* auch t h a t x E )1S.
PROPOSITION 2 . 1 4 :
PROOF:
Let
S b e a n a b s o r b i n g subset o f
6 > 0 such t h a t 1x1 2 6 valued, choose
is
implies
non-trivially n
E
IN s u f f i c i e n t l y b i g
x
E
us. Conversely, l e t
E
1-1 = A n
S C E be
0
is
F
E.
x
E
with
x
Given
E l there AS. S i n c e ( F , I - I ) i s
lAol
such t h a t
E
> 1. T h e n , f o r 11.1
I 2
6 . Hence
a balanced s u b s e t such t h a t given
35
TOPOLOGICAL VECTOR SPACES
x E E
there exists
and / A I z 6 = llij
6 > 0
x = ps, w i t h S
p E F*
s E S.
implies
Hence
x
such t h a t
A
2
1
-1
x = A
-1
E US.
L e t 6 = 1111. Then
IA-'pl.
On t h e o t h e r hand,
11s E S ,
i.e.
x E AS. T h u s
is absorbing. Notice t h a t i n t h e f i r s t p a r t o f t h e proof
(F,1
b a l a n c e d , and i n t h e s e c o n d p a r t ,
*
1)
S
need n o t
be
need n o t b e non-trivi-
a l l y valued.
L e t E be a vecRoh bpaCe V u Q h u non-tkivialLy v a l u e d d i v i b i o n k i n g (F,1 . I ) , and l e t B b e u h u n d a m e n t a l b y h t e m neighbotrhaodn u 6 0 i n E doh borne T V S t t o p o l o g y T. Then
THEOREM 2 . 1 5 :
(i)
doh e a c h
thetre
V E R,
U E R
Qxibtb
huch
that
u + u c v ; (ii) 6otr each
thut
A E F*
wc
and
V E B
Rhetre
exibtb
W E B buCh
AV.
B i b a d u n d a m e n t a l byb.tem 0 6 n e i g h b a h h o o d b a h 0 dvh a T V S t o p o l o g y -i o n E , and b y ( i i ) a b a w e , ( b ) thug doh any A E F*. Then
PROOF:
S t a t e m e n t s ( i )and
operations (x,y)
-+
Conversely, l e t
ties
(a) through
and s o
+
y
and
x
X -1x
-+
a t the origin.
8 be a f i l t e r basis i n
( c ) . Hence e a c h
V
€
E
s a t i s f y i n g proper-
B i s non-empty and balanced,
0 E V.
For each
of
x
( i i ) f o l l o w f r o m t h e c o n t i n u i t y of the
x E El let
F ( x ) be t h e c o l l e c t i o n o f a l l s u b s e t s
E w h i c h c o n t a i n some s e t of t h e form
x +V, with V
E
8. Then
36
PROLLA
8.
F ( 0 ) i s t h e f i l t e r g e n e r a t e d by t h e f i l t e r b a s i s
The
fol-
F ( x ) are t r u e .
lowing p r o p e r t i e s o f t h e f a m i l i e s
Clear.
PROOF:
k
PROOF: T h i s follows from t h e f a c t t h a t
x
E N.
Clear.
PROOF :
PROOF:
N t F(x) ,
aLl
( i i i ) Fak
is a f i l t e r basis.
8
Let
such t h a t
V W
8 be s u c h t h a t
E
+ W
o t h e r hand, i f
C
Let
M
M =
then
y
x
-
+
W.
x
E W.
By ( a ) I c h o o s e
W E €3
C l e a r l y , M E F ( x ) . On t h e Hence
.
N t F(y)
and so
V.
y
x + V C N.
From ( i ) t h r o u g h ( i v ) , i t f o l l o w s t h a t t h e r e e x i s t s a unique topology
T
on E f o r w h i c h , f o r a l l
a l l n e i g h b o r h o o d s of
x E El
From o u r d e f i n i t i o n i t f o l l o w s t h a t borhoods a t (F,
1. 1 ) . (v) Let
exists
U
E 8
0.
(x,y) (xo
+
E
x + y
such t h a t
such t h a t
over
i~ c o n , t i n u u u h .
E x E
U +U
5 1).
8 i s a b a s i s of n e i g h -
I t remains t o prove t h a t ( E , T ) i s a T V S
, yo)
V 6 8
F(x) i s t h e f a m i l y o f
x i n T. ( S e e B o u r b a k i [ 1 2 1 , Chap. I ,
C
be given. L e t
xo + yo V.
Let
x
+ E
N t
V C N.
x + 0
F(xo + y o ) .
There
By ( a ) , t h e r e e x i s t s
U and
y
E
yo + U .
Then
37
TOPOLOGICAL VECTOR SPACES
x - x U
+
and
0
U
YO + U
x
and so
V,
C
- yo
y
belong t o
U. Hence
+
V C N.
xo +
y E
x + y - (xo + yo)
Since x
0
+
E
U E F(xo) and
F (yo), t h e c o n t i n u i t y h a s b e e n p r o v e d .
E
(vi)
(A,x)
Ax
-+
i n co&, tinuou b .
Since Ax
- A 0x0
-
= (A
+ Ao(x - xo) + ( A
AJXo
-
Ao) ( x
-
i t f o l l o w s f r o m ( v ) t h a t t o p r o v e ( v i ) i t i s enough ( v i i ) through
xo E E ,
A
(viii) Foa each
.A
x
(A, x )
PROOF:
(vii) L e t
6 > 0
such t h a t
+
( v i i i ) : If
# 0.
lAoAnl
Ax
/ A />
1. By ( b ) , W
V
I* I).
prove
0 .
x
in c v n t i n u v u b a t
0.
(0,o).
i s absorbing,
Axo
C
AnV
IXI
/ A /< 6-l.
L e t then 0 <
AV.
E
0
exists
there
E V.
< 1,
f o r some
choose
n
Assume
that
IN
that
E
SO
W E 8 . Hence
n
AoA v c v,
is balanced.
DEFINITION 2 . 1 6 : (F,
0
V
i.e.
0 <
AoW C
because
to
cvntinuoub a t
i h
t h e r e i s nothing to prove.
= 0,
X,
Axo
-+
6 implies x
E A-lV,
xo
.+
cvntinuuuo a t
i b
V E 8. S i n c e
Let
5
E F,
V E % .Since
I t follows t h a t
.A
,
( i x ) below:
( v i i ) Fok e a c h
(ix)
xo)
L e t (E,T ) b e a T VS o v e r a v a l u e d d i v i s i o n r i n g
A subset
neighborhood
V
B
C E
is said t o
be
of t h e o r i g i n i n ( E , T )
,
bounded
if,
there is a
f o r every
6 > 0 such
38
PROLLA
1x1 2
that
in
6
implies
F
B C AV.
I t f o l l o w s from P r o p o s i t i o n 2 . 7
t h a t any
is
set
finite
b o u n d e d . C l e a r l y , a n y s u b s e t of a b o u n d e d s e t i s b o u n d e d .
Let
b e a T V S D U C A a v a l u e d d i u h i o n hing (F, / - I ) . I d A and B a f i e b o u n d e d b U b d e t h , n o a h e A + B , A U B, t h e c L o n u h e 0 6 A, and AA d o h each X E F .
PROPOSITION 2 . 1 7 :
PROOF: L e t
b e a n e i g h b o r h o o d of
V
one W such t h a t
and
1x1
NOW
B c AW.
u B
A
implies
6A
and
XW,
6 ) B
implies A
> 6 = max(6
A'
-
AA i s b o u n d e d i f
X
= 0.
xu
for all
1x1
A C XU.
Hence
> 6 , where
PROOF:
I).
6 > 0
0 in
n e i g h b o r h o o d of
in
E
x
0
t K,
K C
n
U
i =I
AV
AW
C
U s u c h t h a t U C V.
XV,
Then
XV
= A U C
is s u c h t h a t
{xi
+
AUbAeRb
2
lh]
6
implies
&Lng
ahe b o u n d e d .
t h e mapping (A,x)
1x1 5
such t h a t
By c o m p a c t n e s s o f K , that
+
and l e t
V be a n o p e n
E.
( 0 , ~ ~ Hence, ) . there e x i s t
0
B C XW
X # 0 , t h e boundedness
If
a compact s u b s e t ,
be
K C E
For each
implies
L e t ( E , T ) be a T VS oueh a u d u e d diuhinion
T h e n a L L compacz
Let
6,
> 0
i s bounded.
PROPOSITION 2 . 1 8 :
(F,1 .
c
dg
2
Ihl
+
and
2.3.
Choose now a c l o s e d 0 - n e i g h b o r h o o d A C
gA > 0
A C
follows f r o m C o r o l l a r y
XA
i n ( E , T ) . Choose a n o t h e r
0
There e x i s t s
W C V.
C XW C AV.
Clearly, of
+
W
1x1 2
such t h a t
(E,T)
6(xo) > 0 6 ( x o ) and
Ax
-+
and
t
-
Vx
x t V
t h e r e are f i n i t e l y many Vx, 1
1.
Let
6 = inf { 6(xl),
is continuous a t 0
neighborhood of
xO
imply
A t t V.
x l r . . - , x n t K such
. . . ,6 ( x n ) 1 .
Let
39
TOPOLOGICAL VECTOR SPACES
and
IA-'l
c
AV.
K
5
6 < 6 ( x i ) . Hence
A
-1
x
(F,1 .
dame waLued d i w i n i a n k i n g
E
Therefore
AV.
( E 2 , ~ 2 ) be t w o T V S
LeZ ( E l , ~ l ) a n d
PROPOSITION 2 . 1 9 :
i.e. x
E V,
I),
and Let ConXinuoud R i n e a k map. T h e n T mapn b o u n d e d
T : El bQ,tb
+
E
oweh t h e be. a bvunded
into
seto. PROOF:
Let
i n (El,-rl) implies
b e a bounded s u b s e t , and l e t
B C El
such t h a t
Then a h u b h e 2
Let F
It1 5 M
duck t h a t PROOF: L e t
(F,I
in
I
S C F
It1 5 1
F
0 < 11-1,
- 1)
S C
T(B)
C
6 > 0
Let
T ( W ) C V.
Then
B C AW.
PROPOSITION 2.20:
then
AT(W)
Hence a n y
s
Therefore
E
(F,
1.
t E F, i n p a r t i c u l a r f o r a l l
f o r all
.A -n
.
c a n b e w r i t t e n i n t h e form
5
IV,~,
and
V = { A E F,
A
M = lpol
be such t h a t
S C F
Let
Hence
DEFINITION 2 . 2 1 :
0 < /A:[
-1
S C V,
1x1 5
i.e.
If
t
Hence
< 6-'.
S i s contained i n
Now
t E S.
6 > 0 s u c h t h a t I A I 6' with Take Xo E F
S
M > 0.
E S.
It/ 5 1).
-
s E S.
s
F,
E
1-1,
Conversely, l e t f o r some
6
i s t r i v i a l l y valued,
)
n sufficiently big
= ll-lovl
Is1
> -
condidek (F,Ibounded,id and doh. aLP t E S.
S C A { t
for
IAI
0
XV.
C
i s n o t t r i v i a l l y valued, choose
> 6,
be such t h a t
of
UA
be bounded. I f
I A o / < 1. F o r
W
id
for a l l
implies
for a l l
V be a neigh-
0 i n ( E 2 , ~ 2 ) . There e x i s t s a neighborhood
b o r h o o d of
1-10 { t
pov
E
F;
with
satisfies
E S
E)
S C AV,
implies be g i v e n .
for all
1x1
> 6.
L e t (E,T) b e a T V S o v e r a t o p o l o g i c a l d i v i s i o n
r i n g ( F , . r F ) . A subset B C E i s Z v t a E R y bounded i f , f o r e a c h n e i g h b o r h o o d V o f 0 i n ( E , T ) t h e r e e x i s t s a f i n i t e s e t Bo C B such t h a t
B
C
Bo + V .
C l e a r l y , a n y compact s e t i s t o t a l l y b o u n d e d , and a n y s u b s e t
PROLLA
40
of a t o t a l l y bounded s e t i s t o t a l l y bounded. A l s o ,
are t w o t o t a l l y bounded s u b s e t s , t h e n
A
u B
and
if A
A and
+
B
are
B
t o t a l l y bounded. PROPOSITION 2 . 2 2 :
The
06
CLUbiLhe
a t o t a & L y bounded hubnek
i h
to-
taQP.y b o u n d e d . B be a t o t a l l y b o u n d e d s u b s e t of a
PROOF: L e t
such t h a t
B
0
b e a n e i g h b o r h o o d of
V
B0
C
+
+
W
W C V.
Let
W.
t h e r e i s some b = b
so
b E
a b
follows t h a t
E
B.
+
Since
+
W
b
W
+
W
a
f
-W Bo
C
PROPOSITION 2 . 2 3 : 1 e Z (El,~l)a n d
+ +
i s a neighborhood b
Bo
+ +
W.
Hence
a -b
W.
Since
-W
B b e a t o t a l l y b o u n d e d s u b s e t of
such t h a t
B C B
0
E
+
T(W)
C V.
Then
W.
b,
W , and
it
-+
o w t h Ahc E2
beth
Let
be
iniu
( E 1 , ~ l )a n d l e t
0 i n ( E 2 , ~ 2 ) . Choose a n e i g h b o r h o o d
0 i n ( E l , ~ l )s u c h t h a t B
of = W,
( E 2 , ~ 2 ) b e Rwa T V S
topoLogicaL d i w i n i o n k i n g ( F , r F ) , and 1 e A T : E l a c o n t i n u u u b L i n e a h map. T h W T mapn ta.taL.Oy b o u n d e d t o t a e e y bounded b e t n .
PROOF: L e t
W
V.
battie
b e a n e i g h b o r h o o d of
. Let
be a f i n i t e s u b s e t such t h a t
C B
belongs to
Bo
(E,r)
i n ( € 3 , ~ ) . Choose a s y m m e t r i c o n e
Bo
such t h a t
B
f
- a + a
Let
T VS
V
W of
Bo b e a f i n i t e s u b s e t of
T(B) C T ( B o )
+
V.
A s a p a r t i c u l a r case, w e h a v e t h e f o l l o w i n g :
COROLLARY 2 . 2 4 :
let
a T V S awck a kupulogical d i w i n i u n R o t a Q L y b o u n d e d , t h e n XB i b R o t a L L y
( E , T ) be
n i n g ( F , T ~ ) .16 B C E i h buunded, do& each X E F.
E v e h y t a t a L L y b o u n d e d h u b n e t oQ a T V S ( E , T ) o v e k a nun-ZhiwiaLLy w a L u c d d i v . i h i o n h i n g (F,I*I) i h b o u n d e d .
PROPOSITION 2 . 2 5 :
PROOF:
Let
B
C
E
be a t o t a l l y bounded s u b s e t .
Let
be
V
a
n e i g h b o r h o o d of t h e o r i g i n . Choose a b a l a n c e d n e i g h b o r h o o d W of 0
such t h a t
that
B C Bo
W
+
+
W.
Bo b e a f i n i t e s u b s e t of
such
W C V.
Let
Since
Bo i s b o u n d e d , t h e r e e x i s t s 6 > 0, which
B
41
TOPOLOG I CAL VECTOR SPACES
w e may assume t o b e s u c h t h a t 6 Ihl 2 6 . Since W i s balanced, Hence
B C Bo
+
+
W C XW
1.
1, w i t h
1x1 2
XW + AW
W C
XW
whenever
implies
W C XW.
Bo 1
6
C
1x1 2
XV, f o r a l l
C
6.
DEFINITION 2 . 2 6 : L e t ( E , T ) b e a T V S o v e r a t o p o l o g i c a l d i v i s i o n
E' the v e c t o r s p a c e o v e r
( F , . r F ) . W e d e n o t e by
ring
F
of
all
c o n t i n u o u s l i n e a r maps f r o m ( E , T ) i n t o ( F , . r F ) . E ' i s c a l l e d t h e
topological d u a l o f (E,T). C l e a r l y , i n a T V S ( E , T ) a l i n e a r map
uous i f , and o n l y i f , PROPOSITION 2 . 2 7 :
f :E
is continuous a t
f
be a T V S
Let ( E J )
+
is contin-
F
0.
U W Q ~ La
nun- L&v~&y
valued
d i v i n i u n h i n g (F,1 . I ) . A l i n e a h map f : E F i n c o n t i n u u u n id, and unLy id, f i n bounded on borne n e i g h b u h h o u d u d 0 . -+
PROOF: I f
f E El,
neighborhood
then
V of
i s c o n t i n u o u s a t 0, a n d t h e r e
f
If ( x ) I 5 M
C o n v e r s e l y , assume t h a t V
/ f (x) 1 5 1
0 such t h a t
i s some n e i g h b o r h o o d o f
x
for a l l
for all
x
is E V.
where
V,
E
0 i n (E,T). By P r o p o s i t i o n 2.20 this
i s e q u i v a l e n t t o s a y t h a t f (V) i s bounded i n ( F , I 1 ) . L e t be g i v e n . L e t 6 > 0 be such t h a t X E F w i t h / A /2 6 plies
S i n c e (F,I 0 <
It( 5 E ) , i s n o n - t r i v i a l l y valued, choose
f i x ) E X W , where
IAI
f(x) E hood o f
w
1)
< 6-I. Then for all 0
E
for all
W = { t E F;
f (x) E X - l W
x
a
for
all
x
x
X E F E
XV. By C o r o l l a r y 2 . 3 , XV
V,
E
> 0 i m-
E V.
with
that
is
i s a neighbor-
in (E,T).
L e t ( E , T ) be a T V S awe& a nun-Rhiui&y valued d i w i n i a n hing ( F ,1 . I ) . A l i n e a h map f : E F i n c u n t i n u u u n id, -1 and u n l q id, i - t d h e h n e l f ( 0 ) in c l o n e d .
PROPOSITION 2 . 2 8 :
-+
PROOF: The c o n d i t i o n i s c l e a r l y n e c e s s a r y , i n (F,
1.1).
Conversely, l e t
i s closed. I f
f : E
f-'(O)=
ous. L e t u s suppose By C o r o l l a r y 2 . 1 1 ,
E,
f-'(O)
+
F
then
since
is closed
{O)
b e a l i n e a r map s u c h t h a t f = 0
and
# E . Choose
Fl(0)
f is c l e a r l y continua E E
with
t h e r e i s a balanced neighborhood
f ( a ) = 1. V
of 0 such
PROLLA
42
that (a
+
V) n f-'(O) / f ( x )I
Indeed, i f
€(a
and V,
+
@. L e t
=
2 1,
then
x
E
-
y =
V.
We c l a i m that
-1 f(x)l x
[
y ) = 0 , a c o n t r a d i c t i o n . Therefore
and by P r o p o s i t i o n 2 . 2 7 ,
DEFINITION 2 . 2 9 :
i t follows t h a t
f
I f ( x ) ( < 1.
belongs
V,
f
to i s bounded
on
i s continuous.
E b e a v e c t o r s p a c e o v e r a v a l u e d division
Let
r i n g (F,I * 1 ) . A s e q u e n c e U = (Un ) of non-empty s u b s e t s o f E i s called a n t h i n g i n E, i f (a)
every
i s b a l a n c e d and a b s o r b i n g :
Un
for a l l
(c)
f o r some
X E F*
X E F*), g i v e n that The s e t the set
Urn C XUn
.
nth
0
m
I I ~
5
3,
for a l l
By Theorem 2 . 1 5 , T~
over
such
E
If
COI,
=
i s m e t r i z a b l e ( B o u r b a k i 1121
A
E
V are strings i n
,
Topologie ggn.,
Chap.
then
E,
U
+
V, U n V
and X U ,
F*, a r e s t r i n g s t o o .
DEFINITION 2 . 3 0 :
( E , T ) be a T V S o v e r a v a l u e d d i v i s i o n ring
Let
(F,1 . 1 ) .
A string
knot
is a
Un
such
no 1, P r o p . 1 ) . U and
If
un
n
n=l
hence f o r a l l
m > n,
U i s t h e f u n d a m e n t a l system
1).
(F,1 .
N(U) =
Ix,
U.
f o r a unique topology
t h a t ( E , r ) i s a T V S over U
then
(and Um,
knot of
8 o f a l l k n o t s of a s t r i n g
of n e i g h b o r h o o d s o f
< 1
there exists
is called the
Un
1x1
with
Un
is called
U = (Un)
-r-neighborhood of
in
0
T-topoLogical,
EXAMPLE 2.31: L e t ( E , T ) b e a T V S o v e r a n o n - t r i v i a l l y
d i v i s i o n r i n g (F,I n e i g h b o r h o o d s of
*1).
0
( c ) of Theorem 2 . 1 5 . logical string
Let
if
each
E.
valued
8 be a fundamental system o f c l o s e d
(a) through w e can choose a T-topo-
in (E,r) satisfying properties For each
Uu = ( U n )
in
U El
E
B
with
U1
= U,
whose k n o t s
"n
43
T O P O L O G I C A L VECTOR SPACES
belong t o
F ( B ) b e t h e s e t o f all s t r i n g s so
Let
€3.
obtained.
F ( B ) has the following properties:
The s e t
F(B)
t h e knots of t h e s t r i n g s belonging t o
(i)
b a s i s of balanced neighborhoods a t (ii) I f
U and
that
W
V belong t o
F(R),
form
for (EJ)
0
there is
a
;
W E F ( B ) such
U n V;
C
DEFINITION 2 . 3 2 : A s e t
F of s t r i n g s i n
E with
property
(ii)
above i s c a l l e d ditecied.
l e t E be a v e c t o h . Apace aveh a d i v i c l i u n h i n g F, and L e i A +. 1x1 be a n o n - f h i v i a L a b a o L u t e v a l u e an F . 7 6 F i h a d i t e c t e d h e t o d h t ) L i M g h i n E , ,then t h e h e t 8 0 6 a L L k n o t h o d a L l h t h i n g n b e l o n g i n g .to F i h a badin 06 b a l a n c e d n e i g h b u k h o o d 4 at 0 doh a [ u n i q u e ) t u p o L o g y T o v e h E , h u c h t h a t ( E , T ) i h a T VS v v e h ( F , 1. 1 ) . The t o p o l o g y T i h h a i d t o be genehated PROPOSITION 2 . 3 3 :
by
F.
B b e t h e s e t of a l l k n o t s of t h e s t r i n g s b e l o n g i n g t o F. S i n c e F i s d i r e c t e d , g i v e n U and V i n F , there is W E F s u c h t h a t W C U n V , and from t h i s i t f o l l o w s t h a t 8 PROOF:
Let
is a f i l t e r basis. Let
string
B . Then
V E
V =
and
W E B
i.e. V
U
n ' belonging t o
U = (UnIn
Let
W
knot
= U
o f some Then
n+l
w + W C V . A
On t h e o t h e r hand, g i v e n with
i s t h e nth
F.
m > n,
such t h a t
W
C
AV.
there is
in
8,
F i n a l l y , e a c h e l e m e n t of
B
E F*,
W = Um
i s b a l a n c e d and a b s o r b i n g , from t h e d e f i n i t i o n o f a s t r i n g i n E. I t r e m a i n s t o a p p l y Theorem 2 . 1 5 .
DEFINITION 2 . 3 4 : L e t ( E J )
(F, 1 .
1).
each knot
A string
Un
u
=
b e a T VS Over a v a l u e d d i v i s i o n r i n g
(unln E m
in
E is called
.r-c&ohed
if
is a closed set i n ( E , T ) .
DEFINITION 2 . 3 5 : A T V S
(EJ)
o v e r a v a l u e d d i v i s i o n r i n g (F,l*l)
44
PROLLA
. r - c l o s e d s t r i n g i n ( E , - c ) i s -c-topo-
i s c a l l e d b a h h e l l e d i f any logical. EXAMPLE 2.36:
Let
E
b e a vector space over
valued d i v i s i o n r i n g ( F , l * 1 ) .
a
non-trivially
The s e t of a L l s t r i n g s
in
c l e a r l y d i r e c t e d , and g e n e r a t e s by P r o p o s i t i o n 2 . 3 3 a
I* I).
such t h a t ( E r r o o ) i s a T V S over ( F ,
-ca i s b a r re 1l e d .
is
topology
Obviously,
(E,T,)
i s t h e f i n e s t T V S topology
N o t i c e t h a t , by Example 2 . 3 1 , T,
on
E
E.
L e t ( E , T ) bc a T V S ouek a n o n - t h i u i a e C g valued d i v i h i o n k i n g (F, I 1 ) h u c h t h a t t h e t o p o L o g i c a C Apace ( E d ) i n a B a i h e h p a c e . Then ( E , T ) i h ba,+~h.eLCed. I n p a h t i c u L a k , a U cornp R e t e rnettrizabee T V S ahe 6ahheRLed.
THEOREM 2 . 3 7 :
-
PROOF: L e t
L/
= (U )
a sequence
Ak
in
'n+l
be a
n ntlN
F* w i t h
lAkl
T - c l o s e d s t r i n g i n ( E , T I . oloose +
as
a,
k
+
Since
00.
each
i s a b s o r b i n g , w e have m
E =
u
k =1
.
XkUn+l
Since ( E , T ) i s a Baire space, and each
is closed,
XkUn+l
at
least o n e of them h a s non-empty i n t e r i o r . By P r o p o s i t i o n 2.2, i f f o l l o w s t h a t Un+l h a s non-empty i n t e r i o r . C a l l i t A. Now
' 'n+l
implies t h a t
0
+
A C Un+l
+ Un
un+l c un
an i n t e r i o r p o i n t of
Un.
Therefore
and t h e s t r i n g
is
T-topological.
0,
U
I n Robertson [ 7 8 ]
,
r
a n d so
0 is
i s a 7-neighborhood
b a r r e l l e d s p a c e s were i n t r o d u c e d
a n o t h e r d e f i n i t i o n ( a n d , i n f a c t , were c a l l e d
of
using
ulthabahuLLed)
.
W e show t h a t t h e two d e f i n i t i o n s are e q u i v a l e n t .
PROPOSITION 2 . 3 8 : L e t ( E , T ) be a T V S ouek a n o n - t h i v i & y
divihion king (F, (a)
(E,T)
I 1).
valued
T h e iaLLowing c o n d i t t i o n n ahe e q u i v a l e n x .
io b a k k e l l e d .
45
TOPOLOGICAL VECTOR SPACES
16 T * in anattheti T V S t u p o L o g g i n E nuch t h a t t h e a t i i g i n hab a dundamentat b y b t e m a d r - c l a h e d n e i g h b a a hoadb, t h e n T* C T.
(b)
PROOF:
(a) * (b): L e t
V b e a -c*-neighborhood o f
a -r-closed T*-neighborhood
W1
of
0 in
such t h a t
E
W i t h o u t l o s s of g e n e r a l i t y w e may a s s u m e t h a t Choose and f i x
X
Suppose t h a t hoods o f
. ,Wn
W1,..
0 such t h a t
k =2,...,n.
hoods o f
1x1
F*, w i t h
E
Wk
By h y p o t h e s i s ,
+
C
V.
W1 i s b a l a n c e d .
a r e balanced T-closed and
Wk C Wk-l
Wk
C
neighborfor a l l
XWk-l
t h e ( b a l a n c e d ) T - c l o s e d -r*-neighbor-
-r*-neighborhood
so t h a t 'n+l
W1
< 1.
0 form a f u n d a m e n t a l s y s t e m f o r
so a b a l a n c e d r - c l o s e d
0 i n E . Choose
+
'n+l
(E,-r*)
of
'n+l
a t the origin, 0 can b e found
n'
and
By i n d u c t i o n , w e h a v e d e f i n e d a T-closed E.
i s b a r r e l l e d , U i s -c-topological.
Since (EJ)
implies t h a t (b)
+ -
string
V i s a T-neighborhood
(a): L e t
F
of
0.
C l e a r l y , F i s d i r e c t e d . By P r o p o s i t i o n 2 . 3 3 , k n o t s of a l l s t r i n g s belonging t o
F
borhoods a t
T*.
0
a r e 7-closed,
f o r a T V S topology
i t f o l l o w s from ( b ) t h a t
Now
Hence
be t h e s e t of a l l T-closed
U = (W ) i n
T*
W1
(F, 1 . 1 ) .
A string
i f every knot
Un
Let
C V
C T.
strings i n
the set
of
E.
all
B o f neighSince a l l elements i n B
form a b a s i s T*
C
T.
Hence t h e k n o t s
of 0 , i . e . o f a n y T-closed s t r i n g a r e T-neighborhoods -r-closed s t r i n g i s T - t o p o l o g i c a l , a n d ( E , T ) i s b a r r e l l e d . DEFINITION 2 . 3 9 :
n
any
( E l T) be a T V S over a valued d i v i s i o n ring
U = (U is E is c a l l e d T-bohnivohaun n n€IN a b s o r b s a l l bounded s e t s i n (E,T); t h a t i s ,
46
PROLLA
€or e a c h
n
such t h a t
IN, g i v e n a bounded s e t B whenever 1x1 6 n
E
.
B C AUn
C
E
there exists 6n>0
C l e a r l y , any r - t o p o l o g i c a l s t r i n g i n ( E l T ) is 7 - h r n i v o r o u s DEFINITION 2 . 4 0 :
(F,
I I) .
.
( E , T ) b e a T V S over a valued d i v i s i o n ring
Let
We s a y t h a t ( E l r) i s b o t n o l o g i c d i f e v e r y r - b o r n i v o r o u s
s t r i n g is r-topological. L e t ( E 1 , ~ l ) and
DEFINITION 2 . 4 1 :
( E 2 , r 2 ) b e two T V S
same v a l u e d d i v i s i o n r i n g ( F , \ - \ ) . A l i n e a r mapping is c a l l e d bounded i f
over the T:
El
+
E2
T maps bounded s e t s i n t o bounded s e t s .
By P r o p o s i t i o n 2 . 1 9 ,
every continuous
linear
mapping
is
bounded. THEOREM 2 . 4 2 :
1 e - t (E,T) be
1 1)
d i u i n i u n K i n g (F,
.
The
a T V S
ouex
a non-ZhiuLaLRq vaRued
dvlLawing ake e q u i v a l e n t :
(a)
(E,r)
(b)
e u e t y b o u n d e d L i n e a h m a p p i n g deljined u n ( E , r ) i n c a n -
i6
boxnvLogical;
tinuoun.
PROOF:
(a)
*
(b): Let
Choose a r * - t o p o l o g i c a l
Clearly, T
-1
(U)
=
because
l o g i c a l . Hence
T-’(W)
continuous a t
0,
string
(T-’(Un))
r-bornivorous,
*
T : (E,T)
+
( E * , T * ) b e a bounded
W be a b a l a n c e d r*-neighborhood o f
mapping. L e t
U = (Un)n E m i n
is a string i n
linear
0 i n (E*,T*). E * w i t h W1=W.
(E,T)
which
is
( a ), T - l ( U ) i s r - t o p o i s a T-neighborhood of 0, and T is T i s b o u n d e d . By
and b e i n g l i n e a r , T
i s continuous.
F be t h e set a l l r-bornivorous s t r i n g s in E . C l e a r l y , F i s d i r e c t e d . By P r o p o s i t i o n 2 . 3 3 , t h e s e t of all k n o t s of a l l s t r i n g s b e l o n g i n g t o F f o r m s a b a s i s 8 o f n e i g h b o r h o o d s a t 0 f o r a T V S t o p o l o g y T*. L e t B C E b e a r-bound& s e t . Then B i s a b s o r b e d by a l l e l e m e n t s i n 8 . Hence B is (b)
r*-bounded.
(a): Let
T h i s shows t h a t t h e i d e n t i t y map f r o m ( E , T )
into
(E,r*) i s bounded. By ( b ) , t h e i d e n t i t y map i s c o n t i n u o u s . T n e r e f o r e , e a c h e l e m e n t of
8 is a r-neighborhood of
0,
and soevery
47
TOPOLOGICAL VECTOR SPACES
F i s r-topological.
element i n
E v e h y m e t f i i z a b l e T V S o v e h a non-,t&ivLaePy d i V i h L o n k i n g (F,1 I ) in bo4noLvgicaL.
COROLLARY 2 . 4 3 :
PROOF:
Let
-
( E , ' I ) b e a m e t r i z a b l e T V S , and l e t
b e a bounded l i n e a r mapping. L e t t h e topology
0 in T
-1
-
= { t E E;
xn
B
=
n E IN
x E T-I(W) n
,
0 i n (E,r).
For each
n
AilUn
Hence
AnW.
€
IN,
0 and t h e r e f o r e c a n n o t b e c o n t a i n e d
and
Now t h e se-
xn $Z T - l ( W ) .
Hence
A-lT(X x) = n n
Tx
is
T(B) C
AW.
par-
and then
E W,
n
and
0
6 > 0 be such t h a t / A / 2 6 i m p l i e s T B) so t h a t / A n [ 2 6. Then T(B) C XnW. In
t i c u l a r , T(A x ) n n
Let
m.
0 i n (E*,T*) such t h a t
i s t o t a l l y bounded.
{Anxn ; n E IN
bounded. L e t Choose
+
i s bounded i n (El?). I n d e e d , hnxn
q u e n c e ( Anxn)
so
E
E defining
F* w i t h ] A , ]
W be a neighborhood of
Choose
(E*,T*)
i s a b a s i s of neighborhoods a t
i s n o t a neighborhood of
-1 ( W ) . T
E
+
d ( t , O ) < 2-"}
Then ( U n f n E T N
i s a neighborhood of
AnlUn
in
n E IN.
(E,T). L e t
(W)
An
T : (Err)
a m e t r i c on
d be
Choose a s e q u e n c e
T.
"n f o r each
vaLued
-1 a c o n t r a d i c t i o n . T h i s c o n t r a d i c t i o n shows that T (W)
must be a n e i g h b o r h o o d of
0
in
( E , T ) , and
T
i s continuous.
r b g e n e r a t e d by t h e s e t F of a l l T-closed s t r i n g s i n (E,'I) i s c a l l e d t h e hfttrong X a p o L o g y of The t o p o l o g y
REMARK 2 . 4 4 :
( E r r ) . Clearly, T T
a linear
b
T'l(V)
Let
and ( E , T ) i s b a r r e l l e d i f , a n d o n l y
C
-rb
if,
is the following: i f
mapping T : ( E J ) + (G,C) i s continuous, then b + ( G I < ) i s c o n t i n u o u s t o o . Indeed, l e t V be a ?-neigh-
b o r h o o d of
U1
,
A n o t h e r i m p o r t a n t p r o p e r t y of
= rb.
T : (E,'I )
at
b
C T
V.
0
in
Then
G.
T
Choose a < - c l o s e d s t r i n g
-1
(U)
i s a 7-closed s t r i n g i n
i s a Tb-neighborhood of B(T)
U in
G starting
E,
and
so
E
such
0 .
be t h e s e t of a l l
T V S topologies
r~ o n
48
PROLLA
that
c 11;
(1)
T
(2)
(E,q)
The s e t
F
is b a r r e l l e d .
F. Then
g e n e r a t e d by
PROOF:
l/ b e a
Let
q-closed
f o r each
T
L
-rL
-rt
Let
be t h e
for all
q
C
t
Since
E.
q t B(T)
.
Thus
is
q,
C
i
(E,q) is barrelled;
q E B ( T ) . Now
for
topology
T VS
6 B(T).
t- c l o s e d s t r i n g i n
is q - t o p o l o g i c a l f o r e a c h
U
E which are n - t o p o l o g i c a l
of a l l s t r i n g s i n
q E B ( T ), is directed.
every
therefore
F , and so
U E
U
is T t - t o p o l o g i c a l .
PROOF: L e t
V
€,-topological
6
b e any string
for all
C q
6-neighborhood o f U = (Un)
q E B(T:.
of
Tt-neighborhood DEFINITION 2 . 4 5 :
0
-
,
Hence
The t o p o l o g y
b a f i t e L L e d t o p o L v g y of
with
U1
C
and,
U1
in
0
a fortiori,
the
V
PROOF:
t
T : (E,T )
v +
be t h e f i n e s t T V S topology
(G,v)
T : ( E , T ~ )+ ( G , v b )
i s continuous.
T : (E,T
5
t
C
v
t
+
(GI()
and so
By ( a ) ,
on (T
t b
v = v
b
,
i.e.
is continuous, 5
T : (E,-rt)
+
(G,C
t
)
C
v
is
v, i . e .
=
)
vb
T
t
.
C v.
barrelled.
v
E
Lineafi
such t h a t
G
i s c o n t i n u o u s , and t h e r e f o r e
this i t f o l l o w s t h a t
a
T.
L e t T : ( E , T ) + ( G I < ) be. a c o n t i n u o u b T : ( E , Tt ) * ( G , E t ) i n c o n t i n u o u n .
Let
is
ac)i)OC,&Ltted
PROPOSITION 2 . 4 6 :
map. T h e n
a
Then U E F, b e c a u s e
V.
is called
T~
Choose
E.
B(C).
is continuous too.
Hence From
Since Hence
49
TOPOLOGICAL VECTOR SPACES DEFINITION 2 . 4 7 :
Clearly,
T'.
T C T'.
t h e same bounded s e t s . I n f a c t ,
Moreover
and
T
generates have
'T
(E,-ra) is a bornological space,
E with t h i s property.
-ra
i s t h e c o a r s e s t b o r n o l o g i c a l t o p o l o g y on
and
E which i s
finer
The s p a c e ( E r r R ) i s c a l l e d t h e a b d o c i a t e d b o f i n o L o g i c a L
T.
Apace o f ( E , T ) . A T VS ( E , T ) i s b o r n o l o g i c a l i f , R T = T .
If
t h e s e t of a l l
is t h e f i n e s t T V S topology
T'
on
than
*I),
i s d i r e c t e d . Hence i t
T-bornivorous s t r i n g s i n ( E J ) a T V S topology
(F,[
( E , T ) i s a TVS over
If
T : (E,T)
T : ( E , Ta )
+
(G,€,)i s
+
a
a continuous
and
only i f ,
l i n e a r mapping, t h e n
( G , < ) i s c o n t i n u o u s too.
PROOF: S i n c e ( C r u x ) i s b a r r e l l e d , i t s u f f i c e s t o show
that
has a fundamental system of p*-closed neighborhoods o f
i s t h e metric d e f i n i n g
p , choose
Un
p
0. I f d
so t h a t
i s a f u n d a m e n t a l s y s t e m of u - c l o s e d b a l a n c e d n e i g h b o r h o o d s of 0 i n (G,p),
s a t i s f y i n g (a) through
For each
n E IN,
p* is b a r r e l l e d ,
x1 E Un+l
with
j
Since ( E x i ) . 3 i =1 y E G.
is p-closed,
x
-
x
E
Tn+2
xj
E
Un+j
j i =1
y E Un
xi
belongs t o
.
x = y.
Indeed, s i n c e
of
Since
Un.
i s p*-topological.
.
+ Tn+,)
=
By i n d u c t i o n , j and x xi i=l
'
i s a p-Cauchy s e q u e n c e ,
Since
W e claim t h a t
~
Then, b y t h e a b o v e r e m a r k (x
d e f i n e a sequence ( x . ) w i t h 7
some
be t h e u*-closure
Tn
the s t r i n g (Tn)n
L e t x E Tn+l. Choose
Let
Ic) o f Theorem 2 . 1 5 .
j
2
i =1
it Un+i
p* C p r
w e can Tn+j+l.
converges C Un
j
( Z
$3.
and
to "n
x i ) converges
i =1
50
to
PROLLA
y i n (G,u*).
x - y
Hence
j EJN.
Now
j
-
x Tk
E
for a l l
x - y # 0. Since
Assume
belongs t o
xi i=l
2
Tn+j+l
for
all
k E IN.
i s a H a u s d o r f f t o p o l o g y , there
p*
i s some p * - n e i g h b o r h o o d V of 0 i n G s u c h t h a t ( x - y + V) n V = @. S i n c e p * C p , t h e r e i s some k t IN s u c h t h a t ( x - y + V ) n U = @. Therefore
x
-
y
9
,
Tk
k
a contradiction.
T h i s e n d s t h e p r o o f of Lemma 2 . 4 8 . L e t (F,1 . 1 )
b e a n o n - t r i v i a l l y v a l u e d d i v i s i o n r i n g . A con-
t i n u o u s l i n e a r mapping Hausdorff
set
TVS
(G,ii)
{(x,Tx); x E El
T from a T V S
over (F,1 . i s closed
p r o d u c t t o p o l o g y . The s o - c a l l e d conditions
1)
i n t h e space T: E
i s c o n t i n u o u s from ( E J )
on
11"
with
with closed
G
-+
the
states
are a s a b o v e ) . N o t i c e
d o r f f T V S ( G , u ) h a s c l o s e d graph i f , and o n l y i f
a Hausdorff T V S topology
i.e. the
from a T V S ( E , T ) i n t o a Haus-
G
-+
E x G
into a
.theotem
cLobed ghaph
u n d e r w h i c h a l i n e a r mapping T: E
1)
h a s a c f o h ~ dg t t a p h ,
g r a p h i s c o n t i n u o u s ( w h e r e (E,T) a n d ( G , p ) t h a t a l i n e a r mapping
o v e r (F,1 .
(EJ)
G such t h a t
,
there exists
p* C
and T
!J
i n t o (G,p*),
L e t u s now p r o v e a " c l o s e d g r a p h t h e o r e m " .
PROOF:
By t h e r e m a r k s p r e c e d i n g t h e s t a t e m e n t
t h e r e e x i s t s a Hausdorff T V S topology
p*
of
on
Theorem 2 . 4 9 , G
such
i ~ *C and T i s c o n t i n u o u s from ( E , T ) into (G,p*). c o n s i d e r t h e a s s o c i a t e d b a r r e l l e d t o p o l o g y ( p * I t = po
.
i s c o n t i n u o u s from ( E , T ) i n t o ( G r p o ) Now, w e have
because
po C
u,
with
is Hausdorff
DEFINITION 2 . 5 0 :
Let
(F,
.
1. I )
r i n g a n d l e t ( E , T ) and (G,v)
po
(Recall th at
T
that
Let us Then T
is barrelled).
b a r r e l l e d . By Lemma 2 . 4 8 ,
po=ii,
be a n o n - t r i v i a l l y valued d i v i s i o n be t w o
TVS
over
i t . Then L ( E ; G )
51
TOPOLOGICAL VECTOR SPACES
denotes into
t h e v e c t o r s p a c e of a l l c o n t i n u o u s l i n e a r
maps
of
G.
DEFINITION 2.51:
G-topologies) : L e t
(
b e a f a m i l y of bound-
ed s u b s e t s o f ( E , T ) c l o s e d u n d e r f i n i t e u n i o n s , a n d l e t a fundamental system of neighborhoods of
6
S E
and
i n (G,v).
0
B
be each
For
let
V E B
Clearly, the set
o f a l l such
8~
= C ( E ; G ) . L e t u s assume
2.15.
E
B
W(S,V) i s a f i l t e r b a s i s o n
s a t i s f i e s (a) t h r o u g h
(c) of
Theorem
Then
# 0,
when U + U C V. S i n c e W(S,AV) = AW(S,V) f o r a l l A s a t i s f i e s ( b ) , and e a c h W(S,hV) i s b a l a n c e d . Let and
W(S,V) b e g i v e n . By P r o p o s i t i o n 2 . 1 9
(G,v)
,
and s o t h e r e e x i s t s
f ( S ) C XV,
i.e.
6 > 0
f ( S ) i s bounded
1X1 2
such t h a t
f E W(S,XV) = A W ( S , V ) .
BG
f E C (E;G)
Hence
6
W(S,V)
in
implies
is
ab-
sorbing. W e h a v e shown t h a t t h e f i l t e r b a s i s
t i e s ( a ) t h r o u g h ( c ) of Theorem 2 . 1 5 , topology over
(F, I
over
TG
1)
L(E;G)
f o r which
86
such t h a t
BG
s a t i s f i e s proper-
and s o t h e r e i s a ( u n i q u e ) ( ~ : ( E ; G ) , T c)
is a
TVS
i s a fundamental system of
neigh-
borhoods. The m o s t i m p o r t a n t e x a m p l e s o f
G-topologies a r e the
fol-
lowing :
i s t h e s e t of a l l l j i n i t e s u b s e t s o f
(a) case (b)
G
TG
i s c a l l e d t h e ,ttopoLogy
E.
In this
0 6 hhnpLe convehgence.
i s t h e f a m i l y of a l l t o t a l l y bounded s u b s e t s
of
i s t h e f a m i l y of a l l compact s u b s e t s o f
In
E.
(c)
G
t h i s case
~6
CUnVehgenCe.
is c a l l e d t h e Ropoeogy
06
E.
compac-t
PROLLA
52
i s t h e f a m i l y of a l l b o u n d e d s e t s of E . I n t h i s case T G i s c a l l e d t h e k o p o e o g y 0 6 unidohm ~ o n v c 4 -
(d)
and w e
g e n c e on bounded b e t b o r t h e b t h O M g t o p o e o g y , w r i t e 1: b ( E ; G ) t o d e n o t e ( & ( E ; G ) , -iG 1 . DEFINITION 2.52: if
A subset
i s s a i d t o be
H C 6(E;G)
i s b o u n d e d i n t h e s p a c e (6( E ; G ) ,
H
In particular, i f
)
G-bounded
.
converH i s p o i n t w i n e bounded or h i m p L y bounded i f
gence, w e say t h a t
it is
T~
is t h e topology of simple
T~
-bounded.
P R O P O S I T I O N 2.53:
LeL
Then k h e doRRu#ing ahe equhm-
H C 6(E;G).
Len.t:
in
(a)
H
(b)
e a c h neighbo4hood V o d abnohbcr e v e h y S E 6 .
(c)
Foh each
6-bounded.
Fa4
,
S E
0 i n (G,v),n If-'(V),
{f(S); f E
U
HI
f
E
bounded
ib
HI i n
(G,v).
PROOF: ( a ) * ( b ) : S i n c e w e h a v e a s s u m e d t h a t is
6 > 0 such t h a t
hV.
Hence (b)
all
* (c):
If
Hence
f t H.
(c) * (a): L e t
1x1 1.
that
1x1
>
6
S C A n {f-'(V);
6
I*1)
implies
H C AW(S,V),
non-
f(S) C
i.e.
f
€
HI
f E C
HI,
then
f(S)
f ( S ) C AV
C
AV f o r
hV. 8 > 0
W(S,V) be given. There e x i s t s
implies
is
f E H}.
S C A n {f-'(V); U{f(S);
(F,
V i s b a l a n c e d . By (a) , t h e r e
t r i v i a l l y v a l u e d w e may a s s u m e t h a t
for all
f
E
H.
such
Hence
H C AW(S,V).
PROPOSITION 2.54:
Let
H C d: ( E ; G ) .
Then t h e doLeowing me eguiva-
RenL: (a)
H
(b)
Foh each neighbohhood
i b eyuicantinuoub.
v 06
0 i n (G,v), n
If-'
(V); f E
HI
53
TOPOLOGICAL VECTOR SPACES
06
in a neighbohhood
0 i n (E,T).
F o h e a c h neighboxhood V u d 0 i n ( G , v ) thehe exints a neighboxhood u 06 0 i n ( E , T ) n u c h t h a t
(c)
PROOF: C l e a r . COROLLARY 2 . 5 5 :
ed d o h e u e h y
An e q u i c a n t i n u o u n A u b h e t o d
d: ( E ; G )
i d
(%bound-
G-topology.
PROPOSITION 2 . 5 6 :
L e t H C E ( E ; G ) b e an e q u i c o n t i n u o u n n u b n e t . The h e n t h i c t i o n n t o H o d t h e doLLowing t o p o l v g i e n a t e Rhe name: (a)
t h e t o p o l o g y o d nimpLe COnUehgence;
(b)
t h e t o p o L o g y ad unidohm c o n u e h g e n c e on t o t u L R y bounded nubnetn.
PROOF:
Let
fo
t a l l y bounded, and to
and
E H,
W(S,V) b e g i v e n , w h e r e
V i s a n e i g h b o r h o o d of
R . Choose a n o t h e r o n e
to-
0 i n ( G , v ) , belonging
such t h a t
R
W E
is
S C E
W + W + W C V ,
a n d , by e q u i c o n t i n u i t y of i n (E,?) s u c h t h a t
f(U)
t a l l y bounded, t h e r e i s S C So
L e t now
with
+
U.
Hence
C
a symmetric neighborhood
U of
for a l l
is
f
a finite
S
E
H.
Since
S
such
subset
0 to-
that
f E W(So,W) i m p l i e s
g E H n [fo
U
W
C
So
+
W ( S o f W ) ] . Then
f E W(So,W). S i n c e
because
H,
i s Symmetric.
g E H and f o r w e see t h a t
f = g
-
Thus
f(S)
C
W
+ W +
W
C
g
V,
= f
0
+
f,
i. e .
54
PROLLA
f E W(S,V).
g belongs to
Therefore
fo
+
W ( S , V ) , and
i f o + w(So,w)l c f o + w ( s , v ) .
H n
L e t (F,1
- 1)
be a non-XhiuiaLlq v a l u e d d i v i s i o n k i n g be. LWu T V S o u e h i ; C u k t h ( G , v ) Haudolr6zj. €1 C X ( E ; G ) i n e q u i c o n t i n u o u b and H1 i 0 t h e c e o s u h e 06 H i n GE (.in t h e ptroduct t o p a ~ u g g ) ,t h e n HI C d:(E;G) and H1 i n eyuicontinuuuh.
LEMMA 2 . 5 7 :
a n d &ex ( E , r ) and
PROOF:
(G,v)
€ t H1, t h e r e e x i s t s a n e t
If
i n t h e product topology.
+
pfa(y)
+
i . l f ( y ) = f (Ax
Af (x)
+
+
n e i g h b o r h o o d of
+
pf (y)
I t follows t h a t
.
in
fa(Ax
is linear. L e t
f
+
liy) = A f a ( x ) +
V be a
+
Xf(x)
v-closed
H i s e q u i c o n t i n u o u s , there exists
Since
G.
H such t h a t fa + f
Since ( G , v ) i s Hausdorff,
py). Hence 0
fa i n
a r-neighborhood U of t h e o r i g i n i n E s u c h t h a t g ( U ) C V f o r a l l g E H . L e t now x E U and f E H1. T h e r e e x i s t s a n e t fa in +
H such t h a t
f ( x ) , and
Therefore tinuous; C
fa
f
+
fa(x) t V
f(U)
C V
i n t h e p r o d u c t t o p o l o g y . Hence f a (x)
f o r all
i n p a r t i c u l a r each
f E HI
f
HI
L(E;G).
L e t (F,1 .
THEOREM 2 . 5 8 :
1)
+
f ( x ) E V, because V i s closed.
implies
and s o
is e q u i c o n is continuous, i . e . H1
H1
be. a non-t&iviaLEy vaeued d i v h i o n 'ting
avid &eA ( E , r i ) and ( G , v ) be bwu T V S o v e & it. 7 6 ( E , q ) i s bakheLLed, t h e n e a c h p o i n X w i s e b o u n d e d s u b s e t H u 6 L(E;G) i s eyuicontinuuu4. PROOF:
V
Let
be a fundamental s y s t e m o f v-closed neighborhoods
of t h e o r i g i n i n For each through
V E V,
V, W
G s a t i s f y i n g ( a ) t h r o u g h (c) o f Theorem let
w
= n {f-'(V);
f E H}.
r u n s through a f i l t e r base
Then, a s
6 in
2.15. runs
V
E which can
be
taken a s a fundamental system of 0-neighborhoods f o r a T V S topology
5 o n E l t h e b o u n d e d n e s s of
H ensuring t h a t each
W
is
a b s o r b i n g . By t h e c o n t i n u i t y o f e a c h Since ( E , q )
f E H, W is Q-closed. i s b a r r e l l e d , by P r o p o s i t i o n 2 . 3 8 w e h a v e 5 C q.
Hence W i s a n e i g h b o r h o o d o f by P r o p o s i t i o n 2 . 5 4 .
0
i n ( E , q ) , and H i s equicontinuous,
55
TOPOLOGICAL VECTOR SPACES
PROOF:
Let
o u s . Now
H =
{fa;
c1
E A}.
f belongs t o t h e c l o s u r e of
t o p o l o g y ) . By Lemma 2 . 5 7 , fa
+
f
By Theorem 2 . 5 8 ,
f
€
d: ( E ; G )
H
in
H i s equicontinu-
GE
( i n the product
, and by P r o p o s i t i o n
u n i f o r m l y on e v e r y t o t a l l y bounded s u b s e t o f
I n I y a h e n [ 34 ]
,
2.56,
(E,rl).
u l t r a b o r n o l o g i c a l (and q u a s i - u l t r a b a r r e l l e d )
spaces w e r e introduced. Following [ l 1
w e dropped
the
prefix
u l t r a . I y a h e n ' s d e f i n i t i o n of an u l t r a b o r n o l o g i c a l space i s t h e
T V S (E,T) is called ultrabornological
following: a
bounded l i n e a r map f r o m ( E J ) [ 3 4 ], p.
D e f i n i t i o n 4.1,
if
every
(see
i n t o any T V S i s c o n t i n u o u s
2 9 8 ) . Our Theorem 2 . 4 2
shows t h a t E f i -
n i t i o n 2.40 a n d I y a h e n I s d e f i n i t i o n a r e e q u i v a l e n t . The o t h e r c l a s s o f T VS i n t r o d u c e d by I y a h e n
is
s p a c e s : t h o s e T V S i n which e v e r y
quasi-ultrabarrelled
that
of
borni-
i s a n e i g h b o r h o o d of t h e o r i g i n . This suggests
vorous u l t r a b a r r e l the following. DEFINITION 2 . 6 0 :
(F,1 .
I).
( E , T ) be a T V S o v e r a v a l u e d d i v i s i o n ring
Let
W e s a y t h a t ( E l ? ) i s quani-ba44eLRed i f e v e r y - r - c l o s e d
7-bornivorous s t r i n g i n
E
i s T-topological.
C l e a r l y , e v e r y b a r r e l l e d and e v e r y b o r n o l o q i c a l
space
is
quasi-barrelled. If
(F,1
(El?)
- 1) ,
i s a T V S o v e r a n o n - t r i v i a l l y valued d i v i s i o n ring
t h e s e t of a l l s t r i n g s w h i c h a r e
T
C . learly,
~
.ra c
Ta ;
Since
T C
-ra.
because
T'
same bounded s e t s a s and only i f ,
T
= T
a
both
-r-closed
Hence i t g e n e r a t e s a T VS
T-bornivorous i s d i r e c t e d . T
and
'T
and
topology
h a v e t h e same bounded sets,
i s t h e f i n e s t T V S t o p o l o g y on E w i t h t h e
.
T.
The s p a c e ( E , T ) i s q u a s i - b a r r e l l e d i f ,
A s i n t h e c a s e of b a r r e l l e d n e s s , o n e c a n d e f i n e a n u n u c i a t e d
56
PROLLA
qiinhi-batrtleLLct1 t a p a e a g q
-ryt
it is the
3 T:
c o a r s e s t quasi-
b a r r e l l e d topology which i s f i n e r t h a t T. To c o n s t r u c t proceeds a s f o l l o w s : gies
on
rl
one
topolo-
c n;
T
(2)
(E,n)
is a q u a s i - b a r r e l l e d .
F of a l l s t r i n g s i n
The s e t
E which a r e n - t o p o l o g i c a l f o r
i s d i r e c t e d . L e t -iYt F . ‘Then T~~ C n f o r a l l
be t h e
Q t QB(-r)
g e n e r a t e d by
PROOF:
.rqt
E such t h a t
(1)
every
Q B ( - r ) b e t h e s e t of a l l T V S
let
U be a s t r i n g i n
Let
Since
topology
T VS
Q E QB(-r).
E which i s c l o s e d and b o r n i v o r o u s
w i t h r e s p e c t to
7qt.
with respe ct to
17, f o r e v e r y
-tqt
C r(
b a r r e l l e d , U is q-topological.
n,
U i s c l o s e d and b o r n i v o r o u s
Since ( E , q ) is quasi-
E QB(T).
Hence
F,
and t h e r e f o r e
U is
t u p u l o g y withi E C 17 5 c T q t . I n p a h t i c u t a h , 7 c Tqt.
aft
U
E
Tqt-topological.
Let
be any <-neighborhood of
V
5-topological
5
cause
C
PROOF: L e t that
c
Tqt
c
n of
,
a
beU,
T’.
Um C V, f o r some
m
E Dl.
the s t r i n g
There i s
E.
II E F , and so T~
-rqt.
Since
T
C
and
17
U is n-topological,
T h i s proves t h a t
C
in
0
closed and 7 - b o r n i v o r o u s ,
r- t Q B ( T ) . Hence
is
(E,q)
for
( E , 7 P ) i s q u a s i - b a r r e l l e d and t h e r e f o r e
Hence
T~~
a
some
and such
Urn i s a rqt-neighborhood
Since ( E , 7 ’ )
space,
C
Choose
0.
U = (Un), which i s
quasi-barrelled,
E.
with
be a Ta-neighborhood of
V
in
0
U1 C V . Then U E F , t Q B ( 7 ) . Hence ’ and f o r t i o r i
11 = ( U n )
r), for a l l
Ta
( C )
string
string
neighborhood
is a
T VS
i-h any
E ~ ~ ( 7 tlzen 1 ,
r)
PROOF:
E
16
(b)
is every of 0 .
bornological T’
E
QB(T).
T’.
The analogue of Theorem 2 . 4 9 i s
true
for
quasi-barrelled
TOPOLOGICAL VECTOR SPACES
57
spaces :
L e t ( E , T ) b e a quani-bahhetted T VS oveh a n o n , t h i w i a e l y w a t u e d d . i v i 4 . i o n h i n g ( F , I I ) T h e n e v e h y bounded f i n e a h t n a p p i n g T 06 E i n t o a compteLe t n e t h i z a b & e T V S ( G , w ) oveh ( F , I I ) , W i t h d o b e d ghaph in c v n . t i n u o u b . THEOREM 2 . 6 1 :
.
-
Theorem 2 . 6 1 i s a c o n s e q u e n c e o f t h e f o l l o w i n g lemmas.
LEMMA 2.62: A b o u n d e d l i n e a h mapping derjined o n a q i i a n i - b m & e d npace. in Meahley con.tiiquouh. LEMMA 2 . 6 3 :
A L i n e a h mapping d e d i n e d o n any T V S
and ~ i L hv d u U
i n a campCete m e , t k i z a b l e T VS, w h i c h i n n e a h t y C o n t i n u o u c S
cloned ghaph,
hab
i 4
and
continuoun.
B e f o r e p r o v i n g t h e a b o v e lemmas l e t u s r e c a l l t h a t a l i n e a r mapping
T : (E,r)+ ( G I < ) i s
<-neighborhood in
E
V
nea4Ly con-tinuuub i f ,
of t h e o r i g i n i n
i s a T-neighborhood
PROOF O F LEMMA 2 . 6 2 :
Let
for
t h e r - c l o s u r e of
GI
c-neighborhood of 0 i n such t h a t
T : (E,7)
G.
+
( G I < ) b e a bounded
For each
T i s bounded, t h e s t r i n g
let
n E IN
Wn
linear V
Choose a 6 - c l o s e d s t r i n g
U1 C V . S i n c e
i s r-bornivorous.
(V)
0.
of
mapping d e f i n e d on a q u a s i - b a r r e l l e d s p a c e ( E , r ) . L e t G
T
every -1
be a
U = (U ) in n (T-l(Un))
b e t h e 7 - c l o s u r e of
-1 ( U n ) .
Then W = (W ) i s a r - c l o s e d 7 - b o r n i v o r o u s s t r i n g i n E. n S i n c e ( E J ) i s q u a s i - b a r r e l l e d , W i s 7 - t o p o l o g i c a l a n d W1 i s a T
T-neighborhood T
-1
of
0.
Since
( V ) i s a r - n e i g h b o r h o o d of
T
-1
(U,)
0 in
T-'(V)
,
E l and
T
C
t h e r - c l o s u r e of
is nearly
con-
tinuous. B e f o r e p r o v i n g Lemma 2 . 6 3 l e t
us introduce the
following
definition: DGFINITION 2 . 6 4 :
A TVS
(GI<)
i s c a l l e d Br-compLete
n e a r l y c o n t i n u o u s l i n e a r mapping
if
every
T I w i t h c l o s e d g r a p h , from a n
a r b i t r a r y T V S ( E , r ) i n t o (G,S;) i s c o n t i n u o u s .
58
PROLLA
Using t h e above d e f i n i t i o n ,
as f o l l o w s :
t h e s t a t e m e n t of Lemma 2.63 reads
cwek~yc o m p t e t c m e t h i z u b e e T V S
Since every complete metrizable
,
Theorem 2 . 3 7 )
,Lid
is
T VS
B
-cvmpLete. r (see
barrelled
Lemma 2 . 6 3 f o l l o w s f r o m Lcmmas2.65 and 2.66
below.
A H a u s d o r f f T V S ( G , E ) i s c a l l e d an indhu-sApace i f , f o r e v e r y coarser I I a u s d o r f f T VS t o p o l o g y 1-1 o n G w e t t have p 3 5 o r , e q u i v a l e n t l y , p t = F; DEFINITION 2.67:
.
PROOF O F LEMMA 2 . 6 5 : let
i~
L e t ( G , C ) be a c o m p l e t e m e t r i z a b l e T VSand
b e a H a u s d o r f f T V S t o p o l o g y on
t h e i d e n t i t y mapping I : ( G , p )
+
h a s a c l o s e d g r a p h a s a m a p p i n g from ( G , p t )
' 1-I.
i-lt
5
C
pt.
By Theorem 2 . 4 9 ,
t h e mapping
By D e f i n i t i o n 2 . 6 7 ,
PROOF OF LEMMA 2 . 6 6 :
infra-s-space. -T
such t h a t
T~
Let C
T.
Let T C
is
I
(7; d e n o t e s t h e
0
0.)
of
(G,C)
,
continuous,
a
be
Hausdorff
since i.e.,
barrelled
b e a H a u s d o r f f T VS t o p o l o g y T V S t o p o l o g y on
as fundamental system of 0-neighborhoods T -neighborhoods
onto
( G I < ) i s an infra-s-space.
(E,-ro)
-r0
It s t i l l
has c l o s e d graph.
(G,C)
5 . Then
coarser t h a n
GI
Since
T~
the
on
E
E w h i c h has
-r-closures of t h e
is barrelled,
-T
i s also
T~
b a r r e l l e d . IIence
( E , T ~ ) is an infra-s-space
On t h e o t h e r h a n d ,
Hence T
= -c0
-T t (T ) = 0
.
T'
0
= T
0
.
Now w e h a v e
-T
0
=
TT 0
--T
and
C T C T~
T
I
C
T
~
i.e.
I t remains t o prove t h e following.
LEMMA 2 . 6 8 :
Let (E,ro) b e a Haundohdd
TVS
duch t h u t , doh
unq
.
59
TOPOLOGICAL VECTOR SPACES
coakbeh Haubdohdd TVS Lopo&ogy have
= T
T
. 0
Then ( E , T ~ )
on
T
b u c h ,that
E
7;
C T,
we
Br-complete.
i d
PROOF: S u p p o s e ( E , - c o ) i s n o t B r - c o m p l e t e . c o n t i n u o u s l i n e a r mapping T : ( G , p )
-+
There e x i s t s a n e a r l y
( E , T ~ )w , ith closed graph,
which i s n o t c o n t i n u o u s . Hence t h e f i n e s t T V S t o p o l o g y T on E , such t h a t T : ( G , p ) gy and
$
T
T
( E r r ) i s c o n t i n u o u s , i s a Hausdorff topolo-
--f
y:
W . e claim t h a t
~
C T.
of 0 i n
E.
Since
i s nearly continuous, T
0.
T
Choose a -ro-neighborhood
BY c o n t i n u i t y o f Thus
mapping t i o n of
T
-1
T-~(u) c
T,
is a
(V)
-1
?J
L e t V b e a "s-neighborhood 0 U of
such t h a t
0
i s a v-neighborhood
(U)
T,
-T
T~
T.
C
i s B -complete,
p-neighborhood of
Thus
T
QED.
r
0
= 7 :
C
V.
of
T-'(V). t h e o r i g i n , and
i s c o n t i n u o u s f r o m ( G , ? J ) i n t o (E,?:).
T
zT
the
By t h e d e f i n i -
a c o n t r a d i c t i o n . Hence ( E , ' r 0 )
The r e s u l t s c o n t a i n e d i n Theorem 2 . 4 9 and 2 . 6 1 a r e known a s the
cLobed
theohem
gtaph
for barrelled
and q u a s i - b a r r e l l e d
e s p a c e s . L e t u s now s t u d y t h e s o - c a l l e d o p e n m a p p i n g t h e o k e m . W
s t a r t with the following d e f i n i t i o n . DEFINITION 2 . 6 9 :
L e t ( E , T ) and
( G , u ) b e two
T V S o v e r t h e same
G v a l u e d d i v i s i o n r i n g (F,I ] ) . A l i n e a r m p T : E i s s a i d t o b e n e a k l y ( o r u l m v b , t ) upen i f for e a c h T - n e i g h b o r h o o d
non-trivially
U
of
0 in
0 in
+
E , t h e u - c l o s u r e of
T(U) is a
1.I-neighborhood
of
G.
PROPOSITION 2 . 7 0 :
(a)
Any lineah m a p p i n g u n t u a bahacLLed Apace
i b neahly open. (b)
Lei
T :E
oh necond catcgohy i n PROOF:
(a)
Let
a T-topological G,
v
=
( T ( U n ) 1.I )
G.
T : (EJ)
b a r r e l l e d space. L e t
be a l i n e a h mapping buch t h a t
G
-+
Then
--f
T i n neatly open.
(G,p) be a
l i n e a r mapping o n t o
U b e a T-neighborhood of
string
U = (U,)
i s a u-closed
T(E) i n
with
U1 C U .
string in
G.
0 in Since
Since
E.
a
Choose
T is onto (G,~.I)
is
60
PROLLA
V
barrelled, (b)
> 1. S i n c e
T i s n e a r l y open.
Hence
U b e a T-neighborhood
Let
1x1
with
i s p-topological.
of
0 in
E.
X
Choose
E
F
i s absorbing, w e have
U
u xku .
E =
l)k Therefore
T ( E ) i s of s e c o n d c a t e g o r y i n
Since
b o r h o o d of
f o r some k .
0
p-neighborhood
of
0 in
By C o r o l l a r y
T(U,) =
n
y
E
T(Un)
for a l l
b a l a n c e d neighborhood o f for a l l
xn
n
0 , so
--t
Hence
L
1. Choose
T(xn)
0 in
xn E Un
0, b e c a u s e
- y E V, because
y E V. T h i s shows t h a t
Xk-T(U) 2.3,
is a
T(U)
p-neigh-
is
a
G.
n) 1
PROOF: L e t
GI
n
{OI.
1. I,
and l e t V b e a c l o s e d and
(G,p). Then so t h a t
( y + V) n T ( U n ) # @ T ( x n ) - y E V. Now
T i s continuous.
V i s closed.
Since
V
i s balanced,
y = 0 , a s ( G r p ) i s a Hausdorff
T VS.
W e a r e now r e a d y t o p r o v e t h e Open Mapping Theorem.
b e a R i n c a k , c o n t i n u o u s , and n e u h l g o p e n m a p p i n g Bkom a cumpCEte m e t h i z a b L e T V S (E,T) into a Huu~duh6d T V S ( G r u ) . T h e n T i b o p e n , i . e . , T maps open n e t b i n t o open THEOREM 2 . 7 2 :
bCk6.
1tL
T
61
TOPOLOG I CAL VECTOR SPACES
PROOF:
I t i s s u f f i c i e n t t o show t h a t f o r some f u n d a m e n t a l system
8 of ?-neighborhoods of
a neighborhood of
in
0
El U
0 i n (G,u). C h o o s e
d e c r e a s i n g sequence and e a c h o p e n , Wn
.
L e t then
.
Wk+2
Un
implies that
B
B so t h a t
R
i s closed. S i n c e
i s a n e i g h b o r h o o d of 0 i n ( G , u ) ,
= T(Un)
If w e show t h a t
complete
E
T(Un) 3 W n + l ,
for all
.
x1
y E Wk+l
Choose
1,
n
=
is
T(U)
is a
(U,)
T is nearly
for a l l n L 1 . is
t h e proof
s u c h t h a t y-T(xl) E
'k+l
By i n d u c t i o n , we c a n d e f i n e a s e q u e n c e (x. ) with x , E 3 J
s+j
and
m
W e c l a i m t h a t t h e p a r t i a l sums o f t h e s e r i e s
form a
Cauchy s e q u e n c e i n
of 0 i n
E.
There e x i s t s
Now,
for all
'=O
p
n
2
E. 0
IN s u c h t h a t f o r a l l
E
'=O
P Xn+i
i
'k+n+i
'k+n T h i s p r o v e s o u r claim. x E E
n > n 0'
0
P i
E xi i =1 I n d e e d , l e t V be a T - n e i g h b o r h o o d
+
'k+n
Since
'k+n
c
v.
'
P
+
'k+n-l (E,T)
'k+n-l
"k+n+i i=l
c v.
i s complete, t h e r e e x i s t s
such t h a t m
x = On t h e o t h e r h a n d ,
Since and
n
Uk
f o r each
2 xi. i=l p 2 0 , w e have seen t h a t
i s c l o s e d , t h i s shows t h a t
1,
x E Uk.
NOW,
for all p 2 0
62
PROLLA
n+w
Letting
p
+
m,
we get Y -
for a l l
n
2
for a l l
n
2
1. By Lemma 2 . 7 1 ,
C
T(Uk) f o r a l l
i.e.
,
Wk+l
THEOREM 2.73:
1. S i n c e
Lct
p e e t e . meLttrizable T Thevi
is continuous, t h i s implies t h a t
T
k
y = T ( x ) . But
1. 1,
T(x) E
T(Uk);
QED.
T be. a c a n . t i n u o u A L i n e a h m a p p i n g 6hom a cmmV S ( E , T ) o n t o ci 6 a t t e l L e d Haudohdd T V S ( G , p ) .
T Xd o p e n .
PROOF:
By p a r t ( a ) o f P r o p o s i t i o n 2 . 7 0 ,
COROLLARY 2.74: T VS.
PROOF:
Let ( E , T )
and
T
i s n e a r l y open.
(G,p) be. Lwu c o m p L e t e rnethizab&e E anto C open.
T h e n any continuous L i n e a t m a p p i n g gtlum
By Theorem 2 . 3 7 ,
(G,p)
is barrelled.
L e t T b e a c o n . t i n u o u d L i n e a h m a p p i n g 6tom a comp l e t e . mctfi-izabLe T V s ( E , T ) o n t o a B a i t e Haundofidd T V S ( G , p . ) . Then T i.) o p e n .
COROLLARY 2 . 7 5 :
PROOF:
By Theorem 2 . 3 7 ,
( G , L J ) i s b a r r e l l e d . O r e l s e , by P r o p o s i -
of s e c o n d c a t e g o r y i n i t s e l f , T i s n e a r l y o p e n and t h e n a p p l y Theorem 2.72. t i o n 2.70,
( b ) , and t h e f a c t t h a t
a Baire s p a c e
COROLLARY 2.76: p
is
L e t (E,T) be. a c o m p L e t e methizablc T V S and Let be any tiausdofi66 T V S Z o p o L o g y on E, w i L h 1-1 C T a n d b u c h
t h a t ( E , p ) i n 4a.kfitLEe.d.
Then
p = 7.
63
TOPOLOGICAL VECTOR SPACES C l e a r l y , t h e C o r o l l a r y a b o v e i s j u s t Lemma
REMARK 2 . 7 7 :
( t h e C l o s e d Graph Theorem)
T h i s shows t h a t Theorem 2 . 4 9
2.48.
a
is
C o r o l l a r y o f t h e Open Mapping Theorem.
L e t T and p be ti^ c o m p l e t e m e t f i i z a b L e T V S t o p v l o g i e n on a vec,toh n p a c e E. 7 A p C T , t h e n = T.
COROLLARY 2 . 7 8 :
L e t ( E , T ) b e a c o m p l e t e me.tkizabLc T VS and L e t b e a Haundok~A T V S . L e t T be a c o n t i n u o u n L i n e a h mapping E i n t o G n u c h t h a t T E ) i n 06 ~ e c o n dcattegohy hn G. T h e n i . n open and o n t o .
THEOREM 2 . 7 9 : (G,p)
06 T
PROOF: By P r o p o s i t i o n 2 . 7 0
i s o p e n by Theorem 2 . 7 2 .
( b ) , T i s almost o p e n .
Hence
Therefore
T
T ( E ) i s o p e n i n G, and t h e r e f o r e
a b s o r b i n g . I t nows f o l l o w s t h a t T i s o n t o , s i n c e T ( E ) i s i n v a r i a n t under scalar m u l t i p l i c a t i o n .
L e t ( E J ) be a c o m p l e t e m e t h h z a b L e T V S and P e t ( G , p ) bc a Haundokdlj T V S . F o k a n y T E L ( E , G ) , e i t h e t T(E) i d ul; Rhe A i h h t c a z e g o h y in G o h T ( E ) = G .
COROLLARY 2.80:
Theorem 2.58 i s t h e e s s e n t i a l i n g r e d i e n t i n t h e p r o o f of the Banach-Steinhaus
Theorem ( 2 . 5 9 ) .
For spaces o v e r
IR
or
c,
W a e l b r o e c k [ 9 5 ] t a k e s i t a s a d e f i n i t i o n o f b a r r e l l e d s p a c e s (of course he
c a l l s them u l t r a b a r r e l l e d s p a c e s ) . S e e D e f i n i t i o n 8 ,
[ 9 5 ] , p a g e 1 0 . H i s P r o p o s i t i o n 5 shows t h a t a n y s p a c e w i t h Banach-Steinhaus
p r o p e r t y i s u l t r a b a r r e l l e d i n t h e s e n s e of
R o b e r t s o n [ 7 8 ] . Hence, i t i s n a t u r a l t o a s k w h e t h e r
the W.
t h e Banach-
S t e i n h a u s p r o p e r t y i m p l i e s b a r r e l l e d n e s s i n t h e s e n s e o f Cefinit i o n 2.35 f o r s p a c e s o v e r v a l u e d d i v i s i o n r i n g s general.
(F,1 .
1)
in
The f o l l o w i n g r e s u l t shows t h a t t h i s i s indeed the case.
THEOREM 2.81:
Let ( E , T ) be a
diwihion hing (F,
I I).
TVS
owch a n o n - t h i w i a & ? y v a l u e d T h e BalLuwing ahe e y u i u a L e n t :
(a)
(E,T) i d bahkelled;
(b)
e a c h p o i n t w i n e bounded n e t H 0 6 c o n t i n u v u d Rincah mappingb 06 ( E , T ) i n t o a T V S ( G , v ) i i 6 equhconRinuoUn.
64
PROLLA
By Theorem 2 . 5 8 , (a) i m p l i e s ( b ) . C o n v e r s e l y , l e t ( E , r ) b e a T V S s a t i s f y i n g c o n d i t i o n (b) above. W e c l a i m t h a t (E,-r) PROOF:
i s b a r r e l l e d . The p r o o f of t h i s c l a i m i s due to Waelbroeck [ 9 5 ] . Let V = ( Vn ) be a -r-closed s t r i n g i n ( E , - r ) . The i d e a o f t h e p r o o f i s t o c o n s t r u c t a T V S (G,w) (W,)
w i t h some fundamental sequence
of v-neighborhoods of t h e o r i g i n , a n d a p o i n t w i s e
family
H of c o n t i n u o u s l i n e a r mappings
:E
+
G
bounded
i s s u c h a way
that
NOW, b y P r o p o s i t i o n 2 . 5 4 ,
c o n t a i n s a -r-neighborhood
Vk
s i n c e , by (b), H i s e q u i c o n t i n u o u s . Thus V
of
0
is a -r-topological
string. Choose and f i x m > n
there exists
fundamental system
ho E F*
I ho I
with
such t h a t
< 1
.
Vm C XoVn
s u c h t h a t g i v e n Vn Choose a n d
B of b a l a n c e d -r-neighborhoods of
a
fix
0 in
E.
R , c h o o s e a T - t o p o l o g i c a l s t r i n g u = (Un) such C IOUn. L e t I b e t h e s e t of a l l such t h a t U1 = U and 'n+l s t r i n g s . The s p a c e G i s t h e a l g e b r a i c d i r e c t sum of t h e f a m i l y { E i ; i E I ) where, f o r e a c h i E I , E . = E . For each k E IN, For each
U E
l e t us define
1
x = ( x1. 11. E I i n by d e f i n i t i o n i f , and o n l y i f Wk C G .
For
xi
G we say t h a t x
E
Wk
i Uk + Vk
E
if
.
i i = (UnInEm
CLAIM:
W
(a) PROOF: L e t if
(Wk) i n a n t t r i n g i n
=
Each
x
wk
E Wk
x = (xi)i t I
.
in and Now
G.
baLanccd and a b d o h b i n g ;
Ih(
Axi
5 1 b e g i v e n . Then 6
Uk
+ Vk
I
if
Xx = ( A x i l i E I
i = (Un)
,
because
65
TOPOLOGICAL VECTOR SPACES
are b a l a n c e d , and
both
Uk
and
that
Wk
is balanced.
L e t now
Vk
x = (xi)i EI
i s absorbing, {xi
1
, ... , x . 1
'k+l Let
PROOF:
i
x
x
Let
x = y
+
z
Uk+l
+
Vk+l
y and
with
z in
i = (U 1 .
if
Wk+l
Hence
n
there exists
Vn
if
E Wk
for
m
.
Then
yi
and
xi
E Uk
+ vk
and
?
for all
x. =O
{ i l ,..., i s } C I .
f o r a f i n i t e set
"k
k E IN.
6otr cc1P
b e g i v e n . Then
E Wm
X -1x
for
implies
+ vk ,
c U;
E Vk
0
=
i
1x1 1. 6
j T h i s shows t h a t
wk ,
c
.
XoVn
x
i s absorbing.
Wk
W e know t h a t g i v e n
PROOF:
vm c
< m.
X -1xi
.
Wk
E
i < j
'k+l
+
belong t o
so
Hence
and so
6
I t follows
c I. Since
{il,...,im1 such t h a t
6 > 0
.
AVk
for a l l
1x1 5
all
C
m
i. = (Ui),
I
there exists
.
Vk
G be g i v e n . W e h av e
i E I, except f o r a f i n i t e set
all
z
in
+
xi E Uk
For e a c h
such
n
i E I,
15 j
2s
that except
w e have
i
xi, E Umj 3 if
=
i i
i
j url+l
= ( C
i.
~~. 1
~
~
i. AoUn3.
we can w r i t e
N ) O W~
Hence
wm
x.
m > n E
This ends t h e proof t h a t Consider i n string
the
G
"m m
implies
i. XoUn3
2
+ XVn a n d 3 and ( c ) is t r u e w i t h
1.
C how,
+
W = (WkIk
T VS t o p o l o g y
v
~
n
+
1, so
Aolx E Wn
X = .A
.
Then
.
is a s t r i n g i n
g e n e r a t e d by t h e
G.
single
W.
For each
i
E
I, let
n1 . :E
+
G
be t h e c a n o n i c a l e m b e d d i n g
66
PROLLA
Notice t h a t f o r e a c h
E . +. G. 1
u; +
Vk
i i = (U )
n n E W ' Since
if that
E d:(E;G).
IT.
1
k
E
c
?Ti-1 (Wk)
i s a -r-neighborhood of
Uk
The f a m i l y
bounded, b e c a u s e e a c h
IN,
H =
-
IT.i I '
0,
w e see
is pointwise
t I}
i s a b s o r b i n g . I n d e e d , g i v e n x E E there
Vk
e x i s t s some 6 > 0 ( d e p e n d i n g on x ) s u c h t h a t I h / > 6 implies x t AV C o n s i d e r now t h e s e t B = { n i ( x ) : i E I } . L e t b B. k '
Then
b
( x ) f o r some
= TT.
i
0
0
if
j
if
io , i f
#
io = ( U n ) n
for each
~~. Thus
C
E G
Hence
XWk
.
bi
c
x t v = Vk
Ut
t AVk C
w + vk ,
.
ri(x)
Thus
because
(X
+
X(Uk
+ Vk),
I t remains to prove t h a t
be such t h a t
W. Then
C
and b . = 0 , 3
x
=
0
7ii(x)
W be a symmetric 7-neighborhood
i E I such t h a t
+ vk
x
B
bi 0
b = (bi)iEI.
k E IN. L e t
i E I. L e t
c u;
E I , and s o
Wk
E
of
E.
x E Uki
a n d so
W) n Vk # @,
Wk , f o r a l l
E
0 in
This
Choose
shows
+ VkC that
i s -r-closed.
Vk
T h i s e n d s t h e p r o o f of Theorem 2 . 8 1 . A similar n o t i o n t o t h a t of a b o r n o l o g i c a l space
(E,r)
over
a t o p o l o g i c a l f i e l d ( F , T ~ w) a s c o n s i d e r e d i n N a c h b i n [ 6 2 ] , 5 8 . To e x p l a i n h i s d e f i n i t i o n w e h a v e t o e x t e n d t h e d e f i n i t i o n o f a bounded s e t g i v e n i n D e f i n i t i o n 2 . 1 6 , f o r s u b s e t s of a T VS ( E , T ) o v e r a v a l u e d d i v i s i o n r i n g ( F , 1 1 ) t o t h e c a s e o f a to-
-
p o l o g i c a l f i e l d ( F , . r F ) , o r more g e n e r a l l y , a t o p o l o g i c a l d i v i s i o n r i n g (F,-rF). DEFINITION 2 . 8 2 :
L e t ( E , T ) b e a T V S o v e r a t o p o l o g i c a l division
ring ( F , T ~ )A . subset -r-neighborhood V
of
0 in
W of
B
C
0 in
F such t h a t
E E,
i s s a i d t o b e b0unde.d i f , g i v e n a
there exists
VB C W.
a
T
F
-neighborhood
Any f i n i t e s e t i s bounded a n d a s u b s e t of a b o u n d e d s e t
is
67
TOPOLOGICAL VECTOR SPACES
bounded. When t h e t o p o l o g y
a subset
is not the discrete
T~
topology,
then
i s b o u n d e d i n (E,T) i f , a n d o n l y i f , f o r e v e r y
B C E
X
of
0
in
When t h e t o p o l o g y
T
i s t h e metric t o p o l o g y d e f i n e d by
?-neighborhood
W
there exists
E,
E F*
such t h a t
XB C W .
A
absolute value coincide.
1x1
+
F
on
then Definition 2.16
F,
Indeed, suppose
B
W a 7-neighborhood
i s bounded i n t h e
E
C
sense
of
there i s sorre
Let
6 > 0
such t h a t Ihj > 6 i m p l i e s B C AW. L e t V = { u G F ; l l ~ l < 6 - l ) . E V, and # 0 , t h e n 1l-l -1 I > 6 , a n d s o B C u - ~ W , i . e .
If
Hence VB
t h e r e i s some
2.82,
Choose
Let
6
Hence
= E
A
-1
-1
.
E
Then
E V
T
> 0
Let
By 2 . 1 6 ,
E.
B C E is b o u n d e d
W be a 7-neighborhood
F- n e i g h b o r h o o d
of
V
of
in
0
0 in
E . By
such t h a t
F
such t h a t
6 > 0
and
0
Conversely, suppose t h a t
C W.
i n t h e s e n s e of 2.82. VB C W.
in
2.82
2.16.
pB C W.
of
and
an
X
1x1
and
-1
> 6 in
i.e.
B C W,
F implies
<
E.
B C XW.
P r o p o s i t i o n 2 . 1 7 and 2 . 1 8 r e m a i n t r u e i f
is a
(EJ)
T VS
over a t o p o l o g i c a l d i v i s i o n r i n g ( F , r F ) a n d " b o u n d e d " i s
meant
from a
T VS
i n t h e s e n s e of D e f i n i t i o n 2 . 8 2 . Also, (E, T
~
e v e r y c o n t i n u o u s l i n e a r map
i )n t o a n o t h e r T V S
(GI T
~
, ) over
T :E
G
--t
t h e same t o p o l o g i c a l divi-
s i o n r i n g (F,rF), maps b o u n d e d s e t s i n t o b o u n d e d s e t s .
For a n y T V S t o p o l o g y
T
on a v e c t o r s p a c e
L ( T ) t h e s e t o f a l l bounded s u b s e t s . C L(-rl).
q i e s on
If E,
{-ri ; i E I } and
T
a T V S t o p o l o g y on
L e t now
8
be
If
i s t h e supremum o f E
T~
i s a non-empty {
C
E,
w e d e n o t e by
r 2 , then
L(-r2)C
f a m i l y of T V S toploT ; ~
i E I ) , then
T
is
and
any
family
of
subsets
of
E,
and
let
68
PROLLA
be t h e f a m i l y of a l l T V S t o p o l o g i e s o n E s u c h that
{
T ; ~i E
B
C L ( T ~ ) S . ince
B C T
E
I}
{@,
E l
i s a T V S topology on
i s bounded, t h e f a m i l y
( R ) = sup
and t h e r e f o r e
i s non-empty.
i E I}
be any T V S t o p o l o g y on
I
Conversely, i f
L(T) 3 B,
then
only i f
T
E.
If
PROPOSITION 2.83:
belongs to
T
on
El
y.
d
r
The topoeogy
{ T ;~i t I]
i b .the d i n e b x
L ( T ) = L(.ri).
it contains
Let
L e t now
Hence
T
id,
topologies
T h i s f a m i l y i s non-empty,
to t h e family
p c
On E A U C ~ and onLg
T V S t t a p o e u g y o n E wkich
7 b e t h e supremum of
belongs
T
E.
{
If
{ T ;~i
T ; ~
L(T)
f
7, t h e n
because
i E I). Now
I}
I I C T V L J C S .
C o n v e r s e l y , if
and
T.
p be a n y T V S t o p o l o g y o n
v p
-
b e t h e f a m i l y of a l l T V S
on E such t h a t T.
i t I}
b e a T V S o v a a totapuLogicd d i u i b i o n
1eZ ( E , T )
hac, .the 4amc b o u n d e d b c . t b a6 PROOF: Let
{Ti;
~ ( 8 ) if,
T C
unique T V S t o p o t a g y R h a i , d o & a n y T V S X v p u k o g y 1-1 o n E , L ( T ) C L ( p ) C
T ( B ) , then
T C
L(T) 2 8.
h i n g (F,T~).T h e h e e x i b t b
id, u
Let
C ~ ( 6 ) Hence . t h e following is true:
T
For a n y T V S t o p o l o g y
(2)
T ; ~
I } . By t h e p r e c e d i n g r e m a r k , w e h a v e
{ T ~ ;i E
L e t now
{
such thatany
E
L(T) = L(r) C L(p).
c
L(V),
and
then
then
TOPOLOG I C A L VECTOR SPACES
69
( N a c h b i n [ 6 2 1 , 5 8 ) : A T VS t o p o l o g y -r o n E i s T = T, where T i s t h e f i n e s t T V S topolo-
DEFINITION 2 . 8 4 :
s a i d t o be b t h U M g i f gy o n
E which h a s t h e s a m e bounded s e t s a s
B of s u b s e t s of
For e v e r y f a m i l y
T.
r(B) is
t h e topology
E,
s t r o n g . I n d e e d , by (1) a b o v e ,
And t h e n b y ( 2 ) a b o v e ;
- c ( B ) i s t h e n c a l l e d t h e AtAong t o p o l o g y genQncLted
The t o p o l o g y
by
8.
THEOREM 2.85: L e t ( E , T ) b e u T V S o v e h u L o p a l a g i c a R d i w i n i o n hing (F,-rF). The I;o&&owing ahe e q u i v a l e n t :
(a)
T
(b)
e w ~ h ybounded L i n e a h m a p p i n g de6ine.d an (E,r) i n c o n t i n -
La a A L h U M g
UUUA
*
PROOF: ( a ) X
C
E
that
u
T : (E, T)
+
(E*,r*)
b e t h e T VS t o p o l o g y on
belongs t o -1
X = T
Lopo&ogy;
.
(b): L e t
mapping. L e t
TVS
p
be a bounded
i f , and o n l y i f t h e r e e x i s t s
(Y).
(which w i l l i m p l y t h a t
p C T
such
Y E T*
C l e a r l y , T i s c o n t i n u o u s from ( E , p ) i n t o ( E * , T * ) . that
linear
E d e f i n e d b y saying that
To p r o v e
i s c o n t i n u o u s from ( E , T )
T
i n t o ( E * , T * ) ) i t s u f f i c e s , by P r o p o s i t i o n 2 . 8 3 and ( a ) , t o show that
L(T)
C L(p).
A C E
L e t then
belong t o
( E , T ) i n t o ( E * , T * ), t h e n A C T-I(B)
T-l(B)
E
,
t o prove t h a t
L(p). Let
d e f i n i t i o n of
p,
(E*,T*) such t h a t
L ( T ) . Since
T(A) = B A E L(p)
T i s bounded
belongs t o
,
it suffices
V be a n open neighborhood of
t h e r e e x i s t s an open V = T-l(W).
Since
L(-r*).
Since
t o show
that
0 i n ( E , p ) . By
neighborhood T(A) = B
from
of
0
in
i s -c*-bounded,
70
PROLLA
t h e r e e x i s t s a TF-neighborhood Hence
UTdl(B)
(b)
=$
i
T-l(W)
(b), T
T
= T,
E* = E ,
REMARK:
0 in
T* =
= L(T), T
L (7
i s continuous, i e. and
of
,
F s u c h t h a t UB C W .
t L(u)
Tel(B)
and
= V,
(a): C o n s i d e r
i d e n t i t y map. S i n c e
U
-
T,
,
as c l a i m e d .
and l e t
T
be
the
i s a bounded l i n e a r map. By
since T c
T c T.
1
T i s always t r u e ,
is strong.
T
Theorem 2 . 8 5 a n d i t s p r o o f a r e d u e t o L .
N a c h b i n (see
8 4 ) . A s a consequence w e
Theorem 8 , Nachbin [ 6 2 1 , p g .
have t h e
following COROLLARY 2 . 8 6 :
divi.5iun king
(b)
Let
(F, I
in a
T
- I).
( E , T ) be
a
T V S
o v e h a n o n - , t h i v i a L L y valued
The dollowing axe cquivaj~en,t:
T VS
h.thong
topohgy.
L e t u s g i v e some e x a m p l e s of s t r o n g T VS t o p o l o g i e s . W e f i r s t
r e c a l l t h a t a T VS ( E , T ) i s c a l l e d locaLLy bounded i f every p o i n t h a s a bounded n e i g h b o r h o o d , o r e q u i v a l e n t l y , i f t h e o r i g i n
has
a bounded n e i g h b o r h o o d .
Id T~ i n n o t t h e d i n c h e t e t v y o e v g y afid ( E , T ) i h a R o c a e L y bounded T V S v v e k ( F , r F ) , t h e n T in a n t h u n g T V S
EXAMPLE 2 . 8 7 :
top0 eogy. PROOF: L e t W
of
T : (E,T)
+
(E*,T*)
b e a T*-neighborhood 0 i n (El
such t h a t
T)
.
Then
AT(V) C W
p o l o g y ) . Now
T(AV)
of
0.
b e a bounded l i n e a r mapping. Let V b e a bounded n e i g h b o r h o o d
Let
T ( V ) i s T*-bounded
toF is not the discrete AV i s a T-neighborhood o f 0 i n E .
(recall that C W,
and
a n d t h e r e e x i s t s h, E F *
T
(Nachbin [ 6 2 ] , p . 85 - 8 6 ) : L e R ( F , T ~ )b e a topeL o g i c a L d i v i b i o n k i n g d u c h t h a t -rF i n n o t $he d i h c k e t e t o p o e o g y and -rF has a c o u n t a b l e BundamentaL n y n t e m v d neighboahaodn at ,the o a i g i n . L e t ( E , T ) b e a T V S O U e h ( F , r F ) . 16 T h a a coun,tizbLe
EXAMPLE 2 . 8 8 :
71
TOPOLOGICAL VECTOR SPACES
at the Ohigin, then
T
i n a
r e s p e c t i v e l y {Un; n = 1 , 2 , 3 ,
...1,
d u n c i a m e n t a l ? n y n t e t n ad n e i g h b o h h o a d n
LopoLogy.
T VS
ntriong
PROOF: L e t
.. I ,
{Vn;n = 1 , 2 , 3 , .
b e a f u n d a m e n t a l s y s t e m o f n e i g h b o r h o o d s a t t h e o r i g i n of ( E J ) , respectively ‘n
’ ‘n+l
that
( F , - r F ) . W i t h o u t loss o f
for a l l
n.
.
L(p) = L (T)
E.
y
b e a n y T VS t o p o l o g y
W e claim t h a t
a s t r o n g topology. in
Let
Let
W
u
n’
v
C
T,
on
assume such
E
w h i c h shows t h a t
T
is
b e any u - n e i g h b o r h o o d of t h e o r i g i n
Suppose t h a t f o r e v e r y
there exists a pair
g e n e r a l i t y w e may
n, with
n’
UnVn
p
i.e.,
W,
un t Un
and
f o r every v
.
E Vn
n
n,
, such
u v 9 W . The s e t A = {v ; n = 1 , 2 , . . I i s -r-bounded. n n n I n d e e d , l e t V b e any ? - n e i g h b o r h o o d of t h e o r i g i n i n E . By con-
that
t i n u i t y a t t h e o r i g i n of t h e mapping ( A , x ) neighborhoods IVlIV2l”’
and
UN
‘VN-l
A
such t h a t
Ax
there
Uvn C V ,
i s -r-bounded,
we can f i n d i n t e g e r
for a l l
1< n 5 N
as c l a i m e d . Hence
k such t h a t
A
UkA C W .
exist
The f i n i t e s e t
UNVN C V.
i s bounded and a n e i g h b o r h o o d
}
be found such t h a t
and so
VN
+
U
of i n
- I.
is
F can
Then
y-bounded,
and
uk v k E W ’
Therefore
which c o n t r a d i c t s t h e a s s u m p t i o n t h a t u v 9 W , f o r a l l n . Hence n n UmVm C W f o r some m . S i n c e T is n o t t h e d i s c r e t e topology, there is
X # 0
a T-neighborhood
with
X E Urn
.
F
Hence
XV
m
C
W
and t h e n
o f t h e o r i g i n . T h i s shows t h a t
C
W
is
T.
NOTES AND REMARKS Most o f t h e c o n t e n t s o f C h a p t e r 2 up t o 2 . 2 8 c a n i n many books o n T V S ; see, f o r e x a m p l e , C h a p t e r I
[111 , o r s e c t i o n s 1 t o 6 o f C h a p t e r I o f S c h a e f e r
of
be
found
Bourbaki
[81]
P r o p o s i t i o n 2.28 r a i s e s t h e f o l l o w i n g q u e s t i o n : w h a t i s t h e w i d e s t c l a s s o f Hausdorff t o p o l o g i c a l d i v i s i o n r i n g s t h a t
can
b e u s e d a s s c a l a r s f o r T VS p r e s e r v i n g t h e p r o p e r t y t h a t l i n e a r f u n c t i o n a l s a r e c o n t i n u o u s i f , and o n l y i f , t h e i r
kernels
are
72
PROLLA
c l o s e d . T h i s w a s a n s w e r e d by N a c h b i n [611. DEFINITION 2.89:
L e t ( F , T ~ b) e a H a u s d o r f f t o p o l o g i c a l d i v i s i o n
r i n g . A Hausdorff topology
w i t h henpeck
06
on
T*
F is s a i d t o be
A Hausdorff t o p o l o g i c a l d i v i s i o n r i n g
d-tXiciiey minimal i f
admiabibLe
( F , T * ) i s a Hausdorff T V S o v e r ( F , T ~ ) .
if
T~
T~
(F,-rF) i s s a i d t o
i s t h e o n l y Hausdorff topology
which i s a d m i s s i b l e w i t h r e s p e c t t o
T
be
on
F
F'
From Theorem 2 o f N a c h b i n [ 6 1 ] and K a p l a n k y ' s c h a r a c t e r i z a -
i t f o l l o w s t h a t any n o n - t r i v i a l l y
t i o n of v a l u e d d i v i s i o n r i n g s
v a l u e d d i v i s i o n r i n g (F,I * I ) i s s t r i c t l y m i n i m a l .
(Nachbin [ 6 1 1 ) : L t . t ( F , T ~ )be. a Haudolrzjd i x p o l o g i c d d i v i n i o n k i n g , T h e dollowing a x t e q u i v a l e n t :
THEOREM 2 . 9 0 :
(a)
(F,-rF)
i n 4 Z h i c R L y mi.nimul;
PROOF: S u p p o s e t h a t ( F , . r F ) i s s t r i c t l y m i n i m a l , and l e t b e a H a u s d o r f f T VS o v e r ( F , T f
functional. If
~ ) .L
is continuous,
i s c l o s e d i n ( F , T ~ ) .I n d e e d , f-l(O)
then
= El
f
a e E
t i n u o u s . Suppose
given a r b i t r a r i l y , l e t saying t h a t
Y
C
belongs t o
T*
to
T
such t h a t
T VS over
f
i s onto
a
linear
{O]
in
(E,T).
# 0.
F t h e topology T*
F,
A
If
f(x)
€
A.
=
conF
defined
T*
is
Hence
i f , and o n l y i f ,
by
f - 1 (Y)
i t i s easy t o see t h a t Y
C
F
E
belonging
f ( X ) = Y . We claim t h a t ( F , T * ) i s a
Hausdorff
( F , . r F ) , i . e . T*
(Alp)
f (a)
= Af(a)-la. Then
i f , and o n l y i f , t h e r e e x i s t s
(see D e f i n i t i o n 2 . 8 2 ) (i)
be
f-l(O) i s closed
i s such t h a t
x
belongs t o
F
T, Since
F
i s i d e n t i c a l l y z e r o and o b v i o u s l y
f ( E ) = F. L e t u s c o n s i d e r o v e r
belongs t o
+
i s c l o s e d because
( F , - r F ) i s H a u s d o r f f by h y p o t h e s i s .
C o n v e r s e l y , assume t h a t t h e k e r n e l If
f : E
et
f-'(O)
(E,T)
+
h
+
.
X
C
i s admissible with r e s p e c t
i n continuoun,
to
TF
73
TOPOLOGICAL VECTOR SPACES
PROOF: L e t ( a , a ) E F x F , and l e t a
+
f3 i n
( F , T * ) . Since
such t h a t f ( a ) =
c1
f
there e x i s t (a,b)
F,
f ( b ) = 8. Now
and
i t follows t h a t
W E T*,
is onto
W be a n open neighborhood of
a
c a n f i n d T-open n e i g h b o r h o o d s A o f a and such t h a t
A
+
B C f-l(W). L e t
(resp. V) i s a
u+vcw.
open
B of b , r e s p e c t i v e l y ,
{O)
Since
the
division
i s r*-closed.
*
(a)
(b)
F.
E
f ( E \ N) = F \ (01,
Now
*
ring
is strictly
(F,-rF)
f
i s c o n t i n u o u s from ( E , T )
minimal,
onto
(E,r)
(F,rF).
( a ) : Suppose ( F , T ~ i) s n o t s t r i c t l y m i n i m a l . L e t
(X,u)
( F , - r F ) . Hence
(F,T*) i n t o ( F , T * ) . P u t t i n g
(F,.rF) x
i d e n t i t y mapping that is
T*
C
rF
.
A
-+
+
Xu
k e r n e l , namely
Hausfrom
see t h a t
we
( 0 3 , because
d e f i n e d by
the
X i s c o n t i n u o u s from ( F , T ~ )i n t o ( F , T * ) ; -r* # T~ , by h y p o t h e s i s , t h e n A X
Since
-+
T*
i s Hausdorff.
a closed
Hence
f a l s e f o r t h e H a u s d o r f f T V S ( F , - r * ) and t h e l i n e a r F
T*
i s continuous
= 1,
i s n o t c o n t i n u o u s from ( F , r * ) i n t o ( F , r F ) , b u t i t h a s
REMARK:
and
(b).
dorff TVS over
+
N =
i s T-open, and
N
be a n o t h e r Hausdorff topology on F s u c h t h a t ( F , T * ) i s a
f : F
that
Since
Now f i s c l e a r l y c o n t i n u o u s as a mapping f r o m .
~
o n t o ( F , T * ) , and s o Hence
6 ) such
U
i s -r*-closed.
{O}
T* = T
i s -r*-open i n
f(E\ N )
V = f ( B ) . Then
a (resp.
f - l ( O ) i f c l o s e d i n ( E , T ) , i t s complement
therefore
so
we
E. Therefore
(i).
PROOF: I f s u f f i c e s t o show t h a t =
x E
b E f - l ( W ) , and since
U = f ( A ) and
n e i g h b o r h o o d of
PROOF: S i m i l a r t o t h a t of
+
a s -r-open i n
f-'(W)
E E
(b)
is
functional
f f X ) = A.
If f o l l o w s f r o m t h e c h a r a c t e r i z a t i o n o f t h e t o p o l o g y
T*
74
PROLLA
g i v e n i n t h e proof of
(a) * ( b ) above, t h a t
only continuous b u t a l s o open, i . e . f (A) A
C
E
belonging t o
T.
f :E
+
is
F
for
all
Hence t h e f o l l o w i n g r e s u l t i s t r u e
(see
Nachbin [ 6 2 1 ,
p. 7 7 ) :
THEOREM 2 . 9 1 :
L e t ( F , T ~ )b e a b t h i c t e y
i s TF-open
not.
minOrid
diuh-
Haundohad
k i n g , and L e i ( E , T ) be a ffaubdohdd T V S v u e h ( F , T ~ ) T. h e n e v e k y n v n - z e k o c o n i i n u v u h L i n e a h duncZionaL f : E + F i b an vpen mapping dkom E o n t o F. biVn
L e t u s now see t h a t
for a strictly
minimal Hausdorff
p o l o g i c a l f i e l d , a " c l o s e d graph theorem" c a n
be
proved
tofor
linear functionals. ( P r o p o s i t i o n 1 9 , N a c h b i n 1 6 2 1 , p . 7 8 ) : LeL (F,TF)
THEOREM 2.92:
divinion king, LcZ ( E , T ) be a Haubdvtr6d T V S v u e k ( F , T ~ )and L e i f : E F b e a LLneatL Aunct i v n a L . T h e dullowing a t e e q u i v a l e n t : b e a b t k i c t l g minima[ Haubdvtdd
-+
(a
f
i b cvntinuvub;
(b
f
hub cLobed g h a p h .
PROOF : S i n c e ( F , T ~ i) s H a u s d o r f f , Conversely, suppose t h a t
(a)
f :E
w i t h c l o s e d graph. L e t us d e f i n e
+
F
E
vector space
E x F, and t h e k e r n e l o f
C o n s i d e r i n g on
E x F
is a +
linear functional by
F
g is a l i n e a r functional onthe g i s t h e graph
t h e product topology, E x F
Hausdorff T V S over (F,-rF). Since
(b) is clear.
g :E x F
for a l l (x,x)
E x F. Clearly,
*
f ( x ) = g(x,O), for a l l
By Theorem 2 . 9 0 ,
x
E
g
of
becomes
f. a
i s continuous. f i s con-
E , o n e sees t h a t
tinuous. S t r i c t l y minimal t o p o l o g i c a l d i v i s i o n r i n g s
are
important
f o r a f u r t h e r r e a s o n , namely t h a t e v e r y f i n i t e - d i m e n s i o n a l s u b -
space of a t o p o l o g i c a l v e c t o r space
over
a
strictly
minimal
75
T O P O L O G I C A L VECTOR SPACES
c o m p l e t e t o p o l o g i c a l d i v i s i o n r i n g i s c l o s e d . T h i s f o l l o w s from Theorem 7 o f Nachbin [61]: THEOREM 2.93:
a Haun-
Nachbin 1 6 1 1 ) : L e t ( F , T ~ b )e
(Theorem 7,
dohdd t o p o C v g i c a L d i v i o i o n k i n g . T h e d o L C o w i n g a h e e q u i u a l e n z : (a)
( F , r F ) i b n t h i c t l y minimae and CompLete.
(b)
Eve4y ~ i n i t e - d i m e n n i o n a B u e c t o h Apace E o u e h F has onCy o n e t o p o l o g y T nuch -that ( E J ) i n a Haundohdd T V S vwek ( F , T ~ ) .
L e t ( F , T ~ b) e a n t t r i c t l ? y m i n i m a L compCete R o p o BogicaC d i v i n i o n h i n g , and L e t ( E , T ) b e a Haubdohda T V S o w c ~ h ( F , - r F ) . T h e n ewehy dinite-dimenniunaC nubnpace 0 6 E i n c C o n e d . COROLLARY 2 . 9 4 :
Since every b a r r e l l e d T V S is quasi-barrelled, t o ask i f t h e c l a s s of b a r r e l l e d spaces i n t h e c l a s s of q u a s i - b a r r e l l e d a d a p t e d from T u r p i n
is
i t i s natural
properly contained
s p a c e s . The f o l l o w i n g
example,
[ 8 9 ] , shows t h a t t h e i n c l u s i o n i s i n d e e d a
proper one. EXAMPLE 2.95:
ring, let
E
Let
(F,
1 1)
vector. L e t
f i n e s t T V S t o p o l o g y on
PROOF: T.
Hence
(E,T)
T~ T
a n o n - t r i v i a l l y valued
division
be t h e s p a c e o f a l l f i n i t e s e q u e n c e s and l e t
be t h e nth-unit
CLAIM I :
be
E f o r which
B
Denote by
T
en the
i s bounded.
i n bofinoBogicaB, h e n c e quani-6ahheLBed.
is stronger than B = T
CLAIM 11: ( E , T )
B = { e n ; n E IN 1 .
.
T
a n d h a s t h e s a m e bounded setsas
in n o t b a 4 h e L l e d .
PROOF: D e f i n e c o n t i n u o u s l i n e a r f u n c t i o n a l s
k 9k(x) =
x
j =1
XjCj
pk : E
+
F
by
76
PROLLA
if
x
=
(6.) 1
where {X.) i s a s e q u e n c e i n
E E;
F with
IX.1
a.
+
3 c o n v e r g e s p o i n t w i s e t o a l i n e a r f u n c t i o n a l P. 7
The s e q u e n c e {P 1 k I f ( E , T ) w e r e b a r r e l l e d , by Theorem 2.59 cp would b e c o n t i n u o u s , and so
v(B) =
would b e bounded, a c o n t r a d i c t i o n .
{A.} 3
I n Definition 2.29, as n
+
a;
( c ) f o l l o w s from (a) and (b),when ln-ll + O
i n p a r t i c u l a r when ( F ,
1. 1 )
i s IR o r C w i t h t h e i r usual
a b s o l u t e v a l u e s . I n d e e d , g i v e n any ho E F*, choose k E IN so that Ik-1 I 5 lXol , a n d t h e n , g i v e n Un , c h o o s e by ( b ) m > n is so b i g t h a t t h e k-fold sum Um + + Um C Un . S i n c e m' -1 -1 -1 b a l a n c e d by ( a ) , x E Urn i m p l i e s k .A x E Urn, and so .A x E UnI
...
which p r o v e s ( c ) The n o t i o n
.
of s t r i n g appeared i n Iyahen
n i t i o n of oupkaDa44eLn and u L t k a b a k k e L ~ (1341 I y a h e n remarks ( [ 3 4 ] , p. 293) t h a t J . W.
[341 i n h i s d e f i -
,
Definition 3.1).
Baker a l s o
considered
t h e n o t i o n of a n u l t r a b a r r e l . According t o 1341
IR or
T V S (E,T) over
sequence ( U such t h a t
n
U1
D e f i n i t i o n 3.1,
exists
a E
t U1
C
B
C is a nup4aba44et i f t h e r e and
'n+l
+
'n+l
for a l l
n '
n
i s c a l l e d a dedining Aequence f o r
ultrabarrel). If
IN.
E
B is closed, it is c a l l e d an uLttaba4ket.
i s bornivorous, B
The
I f each
B.
Un
i s c a l l e d a bornivorous s u p r a b a r r e l
(resp.
B i s a b a l a n c e d convex a n d a b s o r b i n g
subset,
B is a suprabarrel with
then
a balanced s u b s e t B of a
o f b a l a n c e d and a b s o r b i n g non-empty s u b s e t s o f
I f i n addition sequence (Un)
,
2
-n
B = Un
,
n
E
IN,
as a defining
sequence. The n o t i o n o f a non-convex b a r r e l l e d T V S , when C , i s due t o W .
Robertson ( [ 7 8 ], p. 2 4 9 )
, who
F i s IR o r
c a l l e d them UU'UXo v e r IR
6akheLLed. They w e r e d e f i n e d a s f o l l o w s : a T V S ( E , q )
or E,
a:
i s c a l l e d uLtkabatkelLed i f t h e only T V S topologies
i n which t h e r e i s a b a s e o f q - c l o s e d n e i g h b o r h o o d s
o r i g i n , a r e those coarser than
or
q.
P r o p o s i t i o n 2.38,
on
of
the
w i t h F =IR
C , i s due t o I y a h e n ( [ 3 4 1 , Theorem 3 . 1 ) who showed
T V S i s u l t r a b a r r e l l e d i f and o n l y i f e v e r y u l t r a b a r r e l
that is
a a
neighborhood o f t h e o r i g i n .
or
Theorem 2.59 ( B a n a c h - S t e i n h a u s Theorem) , i n t h e case F i s due t o W. R o b e r t s o n ( [ 7 8 ] , Theorem 5 , p . 2 5 0 ) .
C,
=
IR
77
TOPOLOGICAL VECTOR SPACES
Theorem 2 . 4 9 C,
is
( c l o s e d graph Theorem)
When
F = IR o r
,
i n t h e case
Robertson ( [ 7 8 ] , P r o p o s i t i o n
a l s o due t o W. C
Mahowald [ S O ] had
F = IR o r
1 5 , p. 252).
proved t h e f o l l o w i n g
c h a r a c t e r i z a t i o n of b a r r e l l e d l o c a l l y convex spaces: THEOREM 2 . 9 6 :
IR on
C.
L e t ( E , T ) be a R o c a C l y convex Haundoh66 o p a c e o u e h
T h e 6oLLowing a h e e q u i v a l e n t :
(a)
(E,-r)
(b)
d o h e u e h y Banach o p a c e G , t h e 6aLLowing i n Z h u e : a n y L i n e a h m a p p i n g T htrom E i n t o G w h i c h han a c l o n e d g h a p h i n cOM.tiMUOUh.
in b a u e L R e d ;
For a proof see Mahowald [ 5 0 ] , Theorem 2 . 2 ,
p. 1 0 9 .
r e s u l t w a s e x t e n d e d t o non-convex s p a c e s by Iyahen
[34] ,
showed t h a t R o b e r t s o n ' s c l o s e d graph theorem c h a r a c t e r i z e s
This who ul-
t r a b a r r e l l e d spaces: THEOREM 2 . 9 7 :
L e t ( E , T ) be a T V S
UUeh
IR
04
C. T h e ~oUoLLting
atre e q u i u a L e n t : (a) (b)
(E,T) in u l t h a b a h h e L L e d ; h u h euehy c o m p l e t e rneZhic & i n e a h b p a c e G , A h e @?i?vwing t h u e : a n y l i n e a h mapping T 6hom E in-tv G which han a c&oned g h a p h i 0 c o n t i n u o u b .
i d
For a proof see Iyahen [ 3 4 ] , Theorem 3 . 2 , p . 2 9 7 . The n o t i o n of a non-convex
q u a s i - b a r r e l l e d T V S o v e r IR
or
is due t o Iyahen ( [ 3 4 ] , p. 300) , who c a l l e d them quuAi-uUkabahteLLed. Theorem 2 . 6 1 ( c l o s e d graph theorem f o r bounded mapp i n g s ) , i n t h e c a s e F = IR o r C, i s due t o Iyahen ([34] , Theorem 5.1,
p.
301).
I n f a c t , when
F = IR o r
C, more w a s proved. I n d e e d ,
l o c a l l y convex s p a c e s Mahowald ( [ 5 0 ] , Theorem 3 . 1 )
proved
for the
f o l l o w i n g r e l a t i o n between q u a s i - b a r r e l l e d s p a c e s and the c l o s e d graph theorem:
78
PROLLA
THEOREM 2 . 9 8 :
IR
L e t (E,T) be a l o c a l l y c o n v e x Hauddohdd
Apace uvQ.Z
c.. The d o l l a w i n g a t e eyuiwaLenb: quabi-bathellLed;
(a)
(EJ)
(b)
ewetry Banach Apace G , t h e GoLlowing i b t / r u e : a n y bounded l i n e a h mapping 6 t r o m E i n t o G w h i c h hub a c l o n e d gtraph i h c o n t i n u o u d .
ib
60%
For a p r o o f , see Mahowald
[50],
Theorem 3 . 1 , p .
a n a l o g u e f o r non-convex
s p a c e s w a s p r o v e d by I y a h e n :
Let ate. e q u i v a l e n t :
be a T V S
THEOREM 2 . 9 9 :
(E,r)
owP.h
IR
04
109.
Its
C . The d a U o w i n g
(a)
(E,T)
(b)
dux evetry c o m p l e t e m e t h i c l i n e a h Apace G , .the ZjukYowLng i n bhue: a n y bounded l i n e a h mapping d h a m E i n t o G
i b yuabi-u~.thabatrheL&ed;
u h i c h hah a c l o s e d ghaph i n c o n t i n u o u d . F o r a p r o o f see I y a h e n
[ 3 4 J , Theorem 5 . 1 ,
p . 301.
The n o t i o n o f a non-convex b o r n o l o q i c a l T V S , when F i s IR o r C, i s d u e t o I y a h e n ( [ 3 4 ] , D e f i n i t i o n 4 . 1 , p . 298) who called them uL-thabu/rnaLug.icaL. They were d e f i n e d a s f o l l o w s : a T V S ( E , n ) o v e r IR or C is c a l l e d ulthabohnoLogiCaL i f every bounded l i n e a r map from ( E , T ) i n t o a n y T V S o v e r JR o r C i s cont i n u o u s . Our Theorem 2 . 4 2 shows t h a t D e f i n i t i o n 2 . 4 0 and Iyahen's definition a r e equivalent.
The f o l l o w i n g r e s u l t was p r o v e d
by
I y a h e n , and should b e compared w i t h Theorem 2 . 9 9 . THEOREM 2 . 1 0 0 :
Let
(E,T)
b e a T V S oweh IR o h C..
The d u l l o d n g
atre e q u i v a l e n t : (a)
(EJ)
(b)
6 0 4 eve4.y cornpeebe m e t h i c l i n e a h cspace G , t h e do&LoWLng i s t t u e : a n y bounded L i n e a t mapping ( R a m E i n t o G i d cofitinuoun.
i A
ulLttabotrnuLogical;
DEFINITION 2 . 1 :
( F , T ~ b) e a t o p o l o g i c a l d i v i s i o n r i n g . By a
Let
f o p o t u g i c a l ? u e c f o h hpUCe ( T V where on
E
S)
i s a vector s p a c e o v e r
o u e h ( F , f F ) w e mean a p a i r ( E , T ) , F, and T i s a T V S Zopok?ogy
t h a t is
E;
(a)
t h e mapping ( x , y ) + x + y
from ( E J )
( E , T ) to (E,T)
x
i s continuous ;
t h e mapping (X,x) +Ax
(b)
from ( F , T ~ ) ( E , T ) t o ( E , T ) i s
continuous. If
is a
i s d e f i n e d by a n a b s o l u t e v a l u e
T~
1).
(F,1 .
X
IA 1 ,
+
(F,-cF), w e s h a l l say t h a t ( E J )
TVS over
is a
and ( E l T) TVS ~ U W L
C l e a r l y , any t o p o l o g i c a l d i v i s i o n r i n g i s a TVS o v e r
it-
self. If
( E , T ) i s a T V S o v e r (F,rF) a n d
T
# {@,E), w e say t h a t
(E,T) i s a phvpeh T VS o v e r ( F , T ~ ) i; f T = {@,E}, the TVS (EJ) i s c a l l e d i m p h o p e h . Notice t h a t (E,T) i s a n i m p r o p e r T V S i f , and o n l y i f ,
0
C l e a r l y , any non-zero Hausdorff T V S
= E.
(E,T) is a proper TVS.
L e t ( E , T ) b e a T V S o u c h ( F , T ~ )Foh . each xoE E F*, ,the m a p p i n g 3 x x + x and x A0x ate 0
PROPOSITION 2.2: and e a c h
ho
E
homeomvhphihmn a 4
-+
( E , T ) o n X o i.tnek?d.
x
PROOF: The mapping If
ho # 0 , t h e n
too, again
+
x
+
+
Aox
hilx
by D e f i n i t i o n
i s c o n t i n u o u s from is
2.1.
Definition2.1.
i t s i n v e r s e , which i s c o n t i n u o u s In particular,
x
-+
- x
is a
PROLLA
32
homeomorphism. x x + x0 The c o n t i n u i t y of Its i n v e r s e i s t h e mapping x -+ x
follows f r o m D e f i n i t i o n 2 . 1 . = x + (- xo).
-+
- x0
L e t (E,T) b e a T V S o v c 4 (F,rF) and a E E. T h e n 0 i d , and onLy i d , a + V i b u n e i g h b a ~ h o u d ad a. Any b u b b e t V C E i n a n e i g h b o h h o o d 0 6 0 id, and o n L g id, - V i n a neighbntrhood ad 0. Mohe genetraLLy,id X t F*, V C E i b a neighbohhood a d 0 , id und o n L y id, AV i n a ncighbohhaod a6 0 . COROLLARY 2.3:
v c E i n a neighboahood o h
DEFINITION 2.4: A s u b s e t c a l l e d nymme,thic i f
S =
of a vector s p a c e E o v e r
S C E
-
F is
S.
T h e hymmekhic n e i g h b u h h u o d n (E,r) dohm a b a b i b 04 neighbahhuodn a i 0 .
i n a
a6 0
PROPOSITION 2.5:
PROOF: L e t V b e a n e i g h b o r h o o d of 0 i n (E,T). 2 . 3 , - V i s a n e i g h b o r h o o d o f 0 . Hence V n r i c n e i g h b o r h o o d of 0 , w h i c h i s c o n t a i n e d i n
(-
By
T VS
Corollary
V) i s a symmetV.
DEFINITION 2.6: L e t (F,I*I)b e a v a l u e d d i v i s i o n r i n g . A s u b s e t of a v e c t o r s p a c e
S C E
each
x
6 > 0
E l there is a
E
implies
x
E
E over
F is called
a b b o h b i n g i f , for
s u c h t h a t f o r any
h
E
F, ( X ( z 6
AS.
C l e a r l y , i f S i s a b s o r b i n g , and
T 3 S,
then T i s absorbing
too.
PROPOSITION 2.7:
ncighbotrhuod u d
Let ( E , T ) b e a T V S o v e ~(F, 1 0 i n (E,r) i n abborrbing.
V C E
PROOF: L e t
b e a n e i g h b o r h o o d of
b e g i v e n . The mapping
X
t h e r e is
E
6
0 , Then ( A / > 6
= E - ~>
> 0
+
such t h a t
DEFINITION 2.8: L e t ( F ,
1- 1)
Ax Ihl
0
- 1).
Then
evetry
.
i n (E,r) L e t x 6 E
i s c o n t i n u o u s a t t h e o r i g i n . Hence 5 E implies Ax E V. Let
implies
X-lx
E V,
i.e.
x
E
XV.
be a v a l u e d d i v i s i o n r i n g . A s u b s e t
33
TOPOLOGICAL VECTOR SPACES
S
of
E
C
a vector space E over
1x1
for e v e r y Clearly,
S i s b a l a n c e d a n d non-empty,
if
Let
(F, / *
1).
{xu} i n
-
S
S.
then
The cLvnuke o d a bulanced A E X
x
s,
E
and
Axc,
Now
a balanced s u b s e t of a
be
S C E
Let
baLanced i f
AS C
S
< I.
PROPOSITION 2 . 9 : PROOF:
F is called
S
E
1x1
<
and
i h
0 E S.
balanced.
T VS
(EJ)
over
1. Then x i s t h e l i m i t o f a n e t Axc, Ax. T h e r e f o r e Ax E 5, a n d +
i s balanced.
If
{Ax; j h l 5 1, x
then
S C El
balanced set containing
E S }
is t h e
smallest
S. I t i s c a l l e d t h e baLanced h u l l o d S .
C l e a r l y , t h e u n i o n of a n y f a m i l y of b a l a n c e d s e t s a n c e d . Hence, t h e r e e x i s t s for a n y s e t
if
bal-
S C E l a b i g g e s t balanced
s e t K c o n t a i n e d i n S. K i s c a l l e d t h e b a l a n c e d hekneR i t i s non-empty i f , a n d o n l y i f , 0 E S.
06
S,
L e t ( F , I 1 ) b~ a non-ttiviaLLy vaBued d c v h L o n king, and l e t ( E , T ) be a T V S owe& (F,1 . 1 ) . T h e baLanced hefine& 06 aviy tieighbokhood o h 0 i n a n e i g h b o t h o v d 0 6 0.
PROPOSITION 2 . 1 0 :
PROOF: L e t V b e a n e i g h b o r h o o d of 0 i n ( E , T ) . S i n c e ( A , x )
i s c o n t i n u o u s a t (0,O) of
0
(EJ)
Choose of
0,
in
V.
AU =
and
0 < and
such t h a t
1 Xo 1 5
XoW
C
,
there exist
1x1 5
6
+
6 > 0 and neighborhood
x
and
E W
Ax
imply
Xx W
E V.
i s a neighborhood V . Now t h e b a l a n c e d h u l l of AoW i s c o n t a i n e d
Indeed, i f
6 . By C o r o l l a r y 2 . 3 ,
u E AoW, t h e n
u
=
AoW
X 0w ,
with
w
E W.
Hence
hhow E V f o r a n y l h l 5 1, b e c a u s e lhhol = 1x1 * /Xol 5 6 w E W. T h i s shows t h a t A 0 W C K C V , w h e r e K i s t h e b a l -
a n c e d k e r n e l of
V.
COROLLARY 2.11: L e t ( E , r ) b e a T V S awe4 a n u n - , t h i v i a & ? y v a l u e d
d i v i n i v n hing (F, 1 . I ) . T h e balanced neighbohhaadh b a n i d 0 6 neighbohhoodn a t 0.
PROPOSITION 2 . 1 2 :
L e t ( E , T ) be a T V S O v e h
UQ
0 dotm a
( F , T ~ ) T. h e
cloned
34
PROLLA
neighbuhkoodn 0 4
a b a n i o oA ne.i.ghba,thoodn a t
0
L e t V be a n e i g h b o r h o o d of 0 i n ( E l T )
PROOF:
is continuous a t ( O , O ) ,
such t h a t hood of
x
0,
v
Choose v = x
+
W
- W)
(x
E
- w
f o r some
W e h a v e shown t h a t
w
neighborhood
x
k.
E
Since
is a n e i g h b o r h o o d of
W
w E W.
of
W
C
v
Then
n W.
w
.
S i n c e (x,y)
+
t W;
W C V.
Hence
=
v
v
+
0
Hence (x - W) n W # @.
and
x
x+y
is a neighbor-
-W
x.
Hence
+
W of
t h e r e is another neighborhood
L e t now
W C V.
-
0.
do~111
E
x - W implies t h a t
w, t h a t i s
V
x E W + W.
contains the closed
0.
l e t ( E , ? ) b c a T V S t ~ ~ ea hn v n - t h i v i a ! - L y v a e u e d I ) . T h e h e e x i n d n a b u n i n a d neighboahaad6 ad 0 c o n n i a t i n g 06 c l o n e d and b a l a n c e d n e t n , namely x h e s e t 0 4 a U cloned and balanced ncighbothoudn 0 6 0 .
COROLLARY 2 . 1 3 : d i v i n i u n Ring
PROOF: L e t
(F, 1 .
0 i n (E,r). By
V b e a n e i g h b o r h o o d of
t h e r e e x i s t s a closed neighborhood
2.12, V.
By C o r o l l a r y 2 . 1 1 ,
0
contained i n
W.
contained
0
in
t h e r e e x i s t s a balanced neighborhood U of
Hence
0 By P r o p o s i t i o n 2 . 9 ,
W of
Proposition
U
E U C T S C
wc
v.
is balanced; i.e.
b a l a n c e d n e i g h b o r h o o d of
0 contained i n
U
is
a
closed
and
V.
L e t S C E b e a b a l a n c e d o u b s e t 06 a vccxUh n p a c e E a v e h a n a n - t h i u i a l l y valued d i v i n i o n h i n g (F,1 . 1 ) . Then S i n ubnvn.Ling i . 4 , and o n l y id, g i v e n x E E thetie exist5 E F* auch t h a t x E )1S.
PROPOSITION 2 . 1 4 :
PROOF:
Let
S b e a n a b s o r b i n g subset o f
6 > 0 such t h a t 1x1 2 6 valued, choose
is
implies
non-trivially n
E
IN s u f f i c i e n t l y b i g
x
E
us. Conversely, l e t
E
1-1 = A n
S C E be
0
is
F
E.
x
E
with
x
Given
E l there AS. S i n c e ( F , I - I ) i s
lAol
such t h a t
E
> 1. T h e n , f o r 11.1
I 2
6 . Hence
a balanced s u b s e t such t h a t given
35
TOPOLOGICAL VECTOR SPACES
x E E
there exists
and / A I z 6 = llij
6 > 0
x = ps, w i t h S
p E F*
s E S.
implies
Hence
x
such t h a t
A
2
1
-1
x = A
-1
E US.
L e t 6 = 1111. Then
IA-'pl.
On t h e o t h e r hand,
11s E S ,
i.e.
x E AS. T h u s
is absorbing. Notice t h a t i n t h e f i r s t p a r t o f t h e proof
(F,1
b a l a n c e d , and i n t h e s e c o n d p a r t ,
*
1)
S
need n o t
be
need n o t b e non-trivi-
a l l y valued.
L e t E be a vecRoh bpaCe V u Q h u non-tkivialLy v a l u e d d i v i b i o n k i n g (F,1 . I ) , and l e t B b e u h u n d a m e n t a l b y h t e m neighbotrhaodn u 6 0 i n E doh borne T V S t t o p o l o g y T. Then
THEOREM 2 . 1 5 :
(i)
doh e a c h
thetre
V E R,
U E R
Qxibtb
huch
that
u + u c v ; (ii) 6otr each
thut
A E F*
wc
and
V E B
Rhetre
exibtb
W E B buCh
AV.
B i b a d u n d a m e n t a l byb.tem 0 6 n e i g h b a h h o o d b a h 0 dvh a T V S t o p o l o g y -i o n E , and b y ( i i ) a b a w e , ( b ) thug doh any A E F*. Then
PROOF:
S t a t e m e n t s ( i )and
operations (x,y)
-+
Conversely, l e t
ties
(a) through
and s o
+
y
and
x
X -1x
-+
a t the origin.
8 be a f i l t e r basis i n
( c ) . Hence e a c h
V
€
E
s a t i s f y i n g proper-
B i s non-empty and balanced,
0 E V.
For each
of
x
( i i ) f o l l o w f r o m t h e c o n t i n u i t y of the
x E El let
F ( x ) be t h e c o l l e c t i o n o f a l l s u b s e t s
E w h i c h c o n t a i n some s e t of t h e form
x +V, with V
E
8. Then
36
PROLLA
8.
F ( 0 ) i s t h e f i l t e r g e n e r a t e d by t h e f i l t e r b a s i s
The
fol-
F ( x ) are t r u e .
lowing p r o p e r t i e s o f t h e f a m i l i e s
Clear.
PROOF:
k
PROOF: T h i s follows from t h e f a c t t h a t
x
E N.
Clear.
PROOF :
PROOF:
N t F(x) ,
aLl
( i i i ) Fak
is a f i l t e r basis.
8
Let
such t h a t
V W
8 be s u c h t h a t
E
+ W
o t h e r hand, i f
C
Let
M
M =
then
y
x
-
+
W.
x
E W.
By ( a ) I c h o o s e
W E €3
C l e a r l y , M E F ( x ) . On t h e Hence
.
N t F(y)
and so
V.
y
x + V C N.
From ( i ) t h r o u g h ( i v ) , i t f o l l o w s t h a t t h e r e e x i s t s a unique topology
T
on E f o r w h i c h , f o r a l l
a l l n e i g h b o r h o o d s of
x E El
From o u r d e f i n i t i o n i t f o l l o w s t h a t borhoods a t (F,
1. 1 ) . (v) Let
exists
U
E 8
0.
(x,y) (xo
+
E
x + y
such t h a t
such t h a t
over
i~ c o n , t i n u u u h .
E x E
U +U
5 1).
8 i s a b a s i s of n e i g h -
I t remains t o prove t h a t ( E , T ) i s a T V S
, yo)
V 6 8
F(x) i s t h e f a m i l y o f
x i n T. ( S e e B o u r b a k i [ 1 2 1 , Chap. I ,
C
be given. L e t
xo + yo V.
Let
x
+ E
N t
V C N.
x + 0
F(xo + y o ) .
There
By ( a ) , t h e r e e x i s t s
U and
y
E
yo + U .
Then
37
TOPOLOGICAL VECTOR SPACES
x - x U
+
and
0
U
YO + U
x
and so
V,
C
- yo
y
belong t o
U. Hence
+
V C N.
xo +
y E
x + y - (xo + yo)
Since x
0
+
E
U E F(xo) and
F (yo), t h e c o n t i n u i t y h a s b e e n p r o v e d .
E
(vi)
(A,x)
Ax
-+
i n co&, tinuou b .
Since Ax
- A 0x0
-
= (A
+ Ao(x - xo) + ( A
AJXo
-
Ao) ( x
-
i t f o l l o w s f r o m ( v ) t h a t t o p r o v e ( v i ) i t i s enough ( v i i ) through
xo E E ,
A
(viii) Foa each
.A
x
(A, x )
PROOF:
(vii) L e t
6 > 0
such t h a t
+
( v i i i ) : If
# 0.
lAoAnl
Ax
/ A />
1. By ( b ) , W
V
I* I).
prove
0 .
x
in c v n t i n u v u b a t
0.
(0,o).
i s absorbing,
Axo
C
AnV
IXI
/ A /< 6-l.
L e t then 0 <
AV.
E
0
exists
there
E V.
< 1,
f o r some
choose
n
Assume
that
IN
that
E
SO
W E 8 . Hence
n
AoA v c v,
is balanced.
DEFINITION 2 . 1 6 : (F,
0
V
i.e.
0 <
AoW C
because
to
cvntinuoub a t
i h
t h e r e i s nothing to prove.
= 0,
X,
Axo
-+
6 implies x
E A-lV,
xo
.+
cvntinuuuo a t
i b
V E 8. S i n c e
Let
5
E F,
V E % .Since
I t follows t h a t
.A
,
( i x ) below:
( v i i ) Fok e a c h
(ix)
xo)
L e t (E,T ) b e a T VS o v e r a v a l u e d d i v i s i o n r i n g
A subset
neighborhood
V
B
C E
is said t o
be
of t h e o r i g i n i n ( E , T )
,
bounded
if,
there is a
f o r every
6 > 0 such
38
PROLLA
1x1 2
that
in
6
implies
F
B C AV.
I t f o l l o w s from P r o p o s i t i o n 2 . 7
t h a t any
is
set
finite
b o u n d e d . C l e a r l y , a n y s u b s e t of a b o u n d e d s e t i s b o u n d e d .
Let
b e a T V S D U C A a v a l u e d d i u h i o n hing (F, / - I ) . I d A and B a f i e b o u n d e d b U b d e t h , n o a h e A + B , A U B, t h e c L o n u h e 0 6 A, and AA d o h each X E F .
PROPOSITION 2 . 1 7 :
PROOF: L e t
b e a n e i g h b o r h o o d of
V
one W such t h a t
and
1x1
NOW
B c AW.
u B
A
implies
6A
and
XW,
6 ) B
implies A
> 6 = max(6
A'
-
AA i s b o u n d e d i f
X
= 0.
xu
for all
1x1
A C XU.
Hence
> 6 , where
PROOF:
I).
6 > 0
0 in
n e i g h b o r h o o d of
in
E
x
0
t K,
K C
n
U
i =I
AV
AW
C
U s u c h t h a t U C V.
XV,
Then
XV
= A U C
is s u c h t h a t
{xi
+
AUbAeRb
2
lh]
6
implies
&Lng
ahe b o u n d e d .
t h e mapping (A,x)
1x1 5
such t h a t
By c o m p a c t n e s s o f K , that
+
and l e t
V be a n o p e n
E.
( 0 , ~ ~ Hence, ) . there e x i s t
0
B C XW
X # 0 , t h e boundedness
If
a compact s u b s e t ,
be
K C E
For each
implies
L e t ( E , T ) be a T VS oueh a u d u e d diuhinion
T h e n a L L compacz
Let
6,
> 0
i s bounded.
PROPOSITION 2 . 1 8 :
(F,1 .
c
dg
2
Ihl
+
and
2.3.
Choose now a c l o s e d 0 - n e i g h b o r h o o d A C
gA > 0
A C
follows f r o m C o r o l l a r y
XA
i n ( E , T ) . Choose a n o t h e r
0
There e x i s t s
W C V.
C XW C AV.
Clearly, of
+
W
1x1 2
such t h a t
(E,T)
6(xo) > 0 6 ( x o ) and
Ax
-+
and
t
-
Vx
x t V
t h e r e are f i n i t e l y many Vx, 1
1.
Let
6 = inf { 6(xl),
is continuous a t 0
neighborhood of
xO
imply
A t t V.
x l r . . - , x n t K such
. . . ,6 ( x n ) 1 .
Let
39
TOPOLOGICAL VECTOR SPACES
and
IA-'l
c
AV.
K
5
6 < 6 ( x i ) . Hence
A
-1
x
(F,1 .
dame waLued d i w i n i a n k i n g
E
Therefore
AV.
( E 2 , ~ 2 ) be t w o T V S
LeZ ( E l , ~ l ) a n d
PROPOSITION 2 . 1 9 :
i.e. x
E V,
I),
and Let ConXinuoud R i n e a k map. T h e n T mapn b o u n d e d
T : El bQ,tb
+
E
oweh t h e be. a bvunded
into
seto. PROOF:
Let
i n (El,-rl) implies
b e a bounded s u b s e t , and l e t
B C El
such t h a t
Then a h u b h e 2
Let F
It1 5 M
duck t h a t PROOF: L e t
(F,I
in
I
S C F
It1 5 1
F
0 < 11-1,
- 1)
S C
T(B)
C
6 > 0
Let
T ( W ) C V.
Then
B C AW.
PROPOSITION 2.20:
then
AT(W)
Hence a n y
s
Therefore
E
(F,
1.
t E F, i n p a r t i c u l a r f o r a l l
f o r all
.A -n
.
c a n b e w r i t t e n i n t h e form
5
IV,~,
and
V = { A E F,
A
M = lpol
be such t h a t
S C F
Let
Hence
DEFINITION 2 . 2 1 :
0 < /A:[
-1
S C V,
1x1 5
i.e.
If
t
Hence
< 6-'.
S i s contained i n
Now
t E S.
6 > 0 s u c h t h a t I A I 6' with Take Xo E F
S
M > 0.
E S.
It/ 5 1).
-
s E S.
s
F,
E
1-1,
Conversely, l e t f o r some
6
i s t r i v i a l l y valued,
)
n sufficiently big
= ll-lovl
Is1
> -
condidek (F,Ibounded,id and doh. aLP t E S.
S C A { t
for
IAI
0
XV.
C
i s n o t t r i v i a l l y valued, choose
> 6,
be such t h a t
of
UA
be bounded. I f
I A o / < 1. F o r
W
id
for a l l
implies
for a l l
V be a neigh-
0 i n ( E 2 , ~ 2 ) . There e x i s t s a neighborhood
b o r h o o d of
1-10 { t
pov
E
F;
with
satisfies
E S
E)
S C AV,
implies be g i v e n .
for all
1x1
> 6.
L e t (E,T) b e a T V S o v e r a t o p o l o g i c a l d i v i s i o n
r i n g ( F , . r F ) . A subset B C E i s Z v t a E R y bounded i f , f o r e a c h n e i g h b o r h o o d V o f 0 i n ( E , T ) t h e r e e x i s t s a f i n i t e s e t Bo C B such t h a t
B
C
Bo + V .
C l e a r l y , a n y compact s e t i s t o t a l l y b o u n d e d , and a n y s u b s e t
PROLLA
40
of a t o t a l l y bounded s e t i s t o t a l l y bounded. A l s o ,
are t w o t o t a l l y bounded s u b s e t s , t h e n
A
u B
and
if A
A and
+
B
are
B
t o t a l l y bounded. PROPOSITION 2 . 2 2 :
The
06
CLUbiLhe
a t o t a & L y bounded hubnek
i h
to-
taQP.y b o u n d e d . B be a t o t a l l y b o u n d e d s u b s e t of a
PROOF: L e t
such t h a t
B
0
b e a n e i g h b o r h o o d of
V
B0
C
+
+
W
W C V.
Let
W.
t h e r e i s some b = b
so
b E
a b
follows t h a t
E
B.
+
Since
+
W
b
W
+
W
a
f
-W Bo
C
PROPOSITION 2 . 2 3 : 1 e Z (El,~l)a n d
+ +
i s a neighborhood b
Bo
+ +
W.
Hence
a -b
W.
Since
-W
B b e a t o t a l l y b o u n d e d s u b s e t of
such t h a t
B C B
0
E
+
T(W)
C V.
Then
W.
b,
W , and
it
-+
o w t h Ahc E2
beth
Let
be
iniu
( E 1 , ~ l )a n d l e t
0 i n ( E 2 , ~ 2 ) . Choose a n e i g h b o r h o o d
0 i n ( E l , ~ l )s u c h t h a t B
of = W,
( E 2 , ~ 2 ) b e Rwa T V S
topoLogicaL d i w i n i o n k i n g ( F , r F ) , and 1 e A T : E l a c o n t i n u u u b L i n e a h map. T h W T mapn ta.taL.Oy b o u n d e d t o t a e e y bounded b e t n .
PROOF: L e t
W
V.
battie
b e a n e i g h b o r h o o d of
. Let
be a f i n i t e s u b s e t such t h a t
C B
belongs to
Bo
(E,r)
i n ( € 3 , ~ ) . Choose a s y m m e t r i c o n e
Bo
such t h a t
B
f
- a + a
Let
T VS
V
W of
Bo b e a f i n i t e s u b s e t of
T(B) C T ( B o )
+
V.
A s a p a r t i c u l a r case, w e h a v e t h e f o l l o w i n g :
COROLLARY 2 . 2 4 :
let
a T V S awck a kupulogical d i w i n i u n R o t a Q L y b o u n d e d , t h e n XB i b R o t a L L y
( E , T ) be
n i n g ( F , T ~ ) .16 B C E i h buunded, do& each X E F.
E v e h y t a t a L L y b o u n d e d h u b n e t oQ a T V S ( E , T ) o v e k a nun-ZhiwiaLLy w a L u c d d i v . i h i o n h i n g (F,I*I) i h b o u n d e d .
PROPOSITION 2 . 2 5 :
PROOF:
Let
B
C
E
be a t o t a l l y bounded s u b s e t .
Let
be
V
a
n e i g h b o r h o o d of t h e o r i g i n . Choose a b a l a n c e d n e i g h b o r h o o d W of 0
such t h a t
that
B C Bo
W
+
+
W.
Bo b e a f i n i t e s u b s e t of
such
W C V.
Let
Since
Bo i s b o u n d e d , t h e r e e x i s t s 6 > 0, which
B
41
TOPOLOG I CAL VECTOR SPACES
w e may assume t o b e s u c h t h a t 6 Ihl 2 6 . Since W i s balanced, Hence
B C Bo
+
+
W C XW
1.
1, w i t h
1x1 2
XW + AW
W C
XW
whenever
implies
W C XW.
Bo 1
6
C
1x1 2
XV, f o r a l l
C
6.
DEFINITION 2 . 2 6 : L e t ( E , T ) b e a T V S o v e r a t o p o l o g i c a l d i v i s i o n
E' the v e c t o r s p a c e o v e r
( F , . r F ) . W e d e n o t e by
ring
F
of
all
c o n t i n u o u s l i n e a r maps f r o m ( E , T ) i n t o ( F , . r F ) . E ' i s c a l l e d t h e
topological d u a l o f (E,T). C l e a r l y , i n a T V S ( E , T ) a l i n e a r map
uous i f , and o n l y i f , PROPOSITION 2 . 2 7 :
f :E
is continuous a t
f
be a T V S
Let ( E J )
+
is contin-
F
0.
U W Q ~ La
nun- L&v~&y
valued
d i v i n i u n h i n g (F,1 . I ) . A l i n e a h map f : E F i n c o n t i n u u u n id, and unLy id, f i n bounded on borne n e i g h b u h h o u d u d 0 . -+
PROOF: I f
f E El,
neighborhood
then
V of
i s c o n t i n u o u s a t 0, a n d t h e r e
f
If ( x ) I 5 M
C o n v e r s e l y , assume t h a t V
/ f (x) 1 5 1
0 such t h a t
i s some n e i g h b o r h o o d o f
x
for a l l
for all
x
is E V.
where
V,
E
0 i n (E,T). By P r o p o s i t i o n 2.20 this
i s e q u i v a l e n t t o s a y t h a t f (V) i s bounded i n ( F , I 1 ) . L e t be g i v e n . L e t 6 > 0 be such t h a t X E F w i t h / A /2 6 plies
S i n c e (F,I 0 <
It( 5 E ) , i s n o n - t r i v i a l l y valued, choose
f i x ) E X W , where
IAI
f(x) E hood o f
w
1)
< 6-I. Then for all 0
E
for all
W = { t E F;
f (x) E X - l W
x
a
for
all
x
x
X E F E
XV. By C o r o l l a r y 2 . 3 , XV
V,
E
> 0 i m-
E V.
with
that
is
i s a neighbor-
in (E,T).
L e t ( E , T ) be a T V S awe& a nun-Rhiui&y valued d i w i n i a n hing ( F ,1 . I ) . A l i n e a h map f : E F i n c u n t i n u u u n id, -1 and u n l q id, i - t d h e h n e l f ( 0 ) in c l o n e d .
PROPOSITION 2 . 2 8 :
-+
PROOF: The c o n d i t i o n i s c l e a r l y n e c e s s a r y , i n (F,
1.1).
Conversely, l e t
i s closed. I f
f : E
f-'(O)=
ous. L e t u s suppose By C o r o l l a r y 2 . 1 1 ,
E,
f-'(O)
+
F
then
since
is closed
{O)
b e a l i n e a r map s u c h t h a t f = 0
and
# E . Choose
Fl(0)
f is c l e a r l y continua E E
with
t h e r e i s a balanced neighborhood
f ( a ) = 1. V
of 0 such
PROLLA
42
that (a
+
V) n f-'(O) / f ( x )I
Indeed, i f
€(a
and V,
+
@. L e t
=
2 1,
then
x
E
-
y =
V.
We c l a i m that
-1 f(x)l x
[
y ) = 0 , a c o n t r a d i c t i o n . Therefore
and by P r o p o s i t i o n 2 . 2 7 ,
DEFINITION 2 . 2 9 :
i t follows t h a t
f
I f ( x ) ( < 1.
belongs
V,
f
to i s bounded
on
i s continuous.
E b e a v e c t o r s p a c e o v e r a v a l u e d division
Let
r i n g (F,I * 1 ) . A s e q u e n c e U = (Un ) of non-empty s u b s e t s o f E i s called a n t h i n g i n E, i f (a)
every
i s b a l a n c e d and a b s o r b i n g :
Un
for a l l
(c)
f o r some
X E F*
X E F*), g i v e n that The s e t the set
Urn C XUn
.
nth
0
m
I I ~
5
3,
for a l l
By Theorem 2 . 1 5 , T~
over
such
E
If
COI,
=
i s m e t r i z a b l e ( B o u r b a k i 1121
A
E
V are strings i n
,
Topologie ggn.,
Chap.
then
E,
U
+
V, U n V
and X U ,
F*, a r e s t r i n g s t o o .
DEFINITION 2 . 3 0 :
( E , T ) be a T V S o v e r a v a l u e d d i v i s i o n ring
Let
(F,1 . 1 ) .
A string
knot
is a
Un
such
no 1, P r o p . 1 ) . U and
If
un
n
n=l
hence f o r a l l
m > n,
U i s t h e f u n d a m e n t a l system
1).
(F,1 .
N(U) =
Ix,
U.
f o r a unique topology
t h a t ( E , r ) i s a T V S over U
then
(and Um,
knot of
8 o f a l l k n o t s of a s t r i n g
of n e i g h b o r h o o d s o f
< 1
there exists
is called the
Un
1x1
with
Un
is called
U = (Un)
-r-neighborhood of
in
0
T-topoLogical,
EXAMPLE 2.31: L e t ( E , T ) b e a T V S o v e r a n o n - t r i v i a l l y
d i v i s i o n r i n g (F,I n e i g h b o r h o o d s of
*1).
0
( c ) of Theorem 2 . 1 5 . logical string
Let
if
each
E.
valued
8 be a fundamental system o f c l o s e d
(a) through w e can choose a T-topo-
in (E,r) satisfying properties For each
Uu = ( U n )
in
U El
E
B
with
U1
= U,
whose k n o t s
"n
43
T O P O L O G I C A L VECTOR SPACES
belong t o
F ( B ) b e t h e s e t o f all s t r i n g s so
Let
€3.
obtained.
F ( B ) has the following properties:
The s e t
F(B)
t h e knots of t h e s t r i n g s belonging t o
(i)
b a s i s of balanced neighborhoods a t (ii) I f
U and
that
W
V belong t o
F(R),
form
for (EJ)
0
there is
a
;
W E F ( B ) such
U n V;
C
DEFINITION 2 . 3 2 : A s e t
F of s t r i n g s i n
E with
property
(ii)
above i s c a l l e d ditecied.
l e t E be a v e c t o h . Apace aveh a d i v i c l i u n h i n g F, and L e i A +. 1x1 be a n o n - f h i v i a L a b a o L u t e v a l u e an F . 7 6 F i h a d i t e c t e d h e t o d h t ) L i M g h i n E , ,then t h e h e t 8 0 6 a L L k n o t h o d a L l h t h i n g n b e l o n g i n g .to F i h a badin 06 b a l a n c e d n e i g h b u k h o o d 4 at 0 doh a [ u n i q u e ) t u p o L o g y T o v e h E , h u c h t h a t ( E , T ) i h a T VS v v e h ( F , 1. 1 ) . The t o p o l o g y T i h h a i d t o be genehated PROPOSITION 2 . 3 3 :
by
F.
B b e t h e s e t of a l l k n o t s of t h e s t r i n g s b e l o n g i n g t o F. S i n c e F i s d i r e c t e d , g i v e n U and V i n F , there is W E F s u c h t h a t W C U n V , and from t h i s i t f o l l o w s t h a t 8 PROOF:
Let
is a f i l t e r basis. Let
string
B . Then
V E
V =
and
W E B
i.e. V
U
n ' belonging t o
U = (UnIn
Let
W
knot
= U
o f some Then
n+l
w + W C V . A
On t h e o t h e r hand, g i v e n with
i s t h e nth
F.
m > n,
such t h a t
W
C
AV.
there is
in
8,
F i n a l l y , e a c h e l e m e n t of
B
E F*,
W = Um
i s b a l a n c e d and a b s o r b i n g , from t h e d e f i n i t i o n o f a s t r i n g i n E. I t r e m a i n s t o a p p l y Theorem 2 . 1 5 .
DEFINITION 2 . 3 4 : L e t ( E J )
(F, 1 .
1).
each knot
A string
Un
u
=
b e a T VS Over a v a l u e d d i v i s i o n r i n g
(unln E m
in
E is called
.r-c&ohed
if
is a closed set i n ( E , T ) .
DEFINITION 2 . 3 5 : A T V S
(EJ)
o v e r a v a l u e d d i v i s i o n r i n g (F,l*l)
44
PROLLA
. r - c l o s e d s t r i n g i n ( E , - c ) i s -c-topo-
i s c a l l e d b a h h e l l e d i f any logical. EXAMPLE 2.36:
Let
E
b e a vector space over
valued d i v i s i o n r i n g ( F , l * 1 ) .
a
non-trivially
The s e t of a L l s t r i n g s
in
c l e a r l y d i r e c t e d , and g e n e r a t e s by P r o p o s i t i o n 2 . 3 3 a
I* I).
such t h a t ( E r r o o ) i s a T V S over ( F ,
-ca i s b a r re 1l e d .
is
topology
Obviously,
(E,T,)
i s t h e f i n e s t T V S topology
N o t i c e t h a t , by Example 2 . 3 1 , T,
on
E
E.
L e t ( E , T ) bc a T V S ouek a n o n - t h i u i a e C g valued d i v i h i o n k i n g (F, I 1 ) h u c h t h a t t h e t o p o L o g i c a C Apace ( E d ) i n a B a i h e h p a c e . Then ( E , T ) i h ba,+~h.eLCed. I n p a h t i c u L a k , a U cornp R e t e rnettrizabee T V S ahe 6ahheRLed.
THEOREM 2 . 3 7 :
-
PROOF: L e t
L/
= (U )
a sequence
Ak
in
'n+l
be a
n ntlN
F* w i t h
lAkl
T - c l o s e d s t r i n g i n ( E , T I . oloose +
as
a,
k
+
Since
00.
each
i s a b s o r b i n g , w e have m
E =
u
k =1
.
XkUn+l
Since ( E , T ) i s a Baire space, and each
is closed,
XkUn+l
at
least o n e of them h a s non-empty i n t e r i o r . By P r o p o s i t i o n 2.2, i f f o l l o w s t h a t Un+l h a s non-empty i n t e r i o r . C a l l i t A. Now
' 'n+l
implies t h a t
0
+
A C Un+l
+ Un
un+l c un
an i n t e r i o r p o i n t of
Un.
Therefore
and t h e s t r i n g
is
T-topological.
0,
U
I n Robertson [ 7 8 ]
,
r
a n d so
0 is
i s a 7-neighborhood
b a r r e l l e d s p a c e s were i n t r o d u c e d
a n o t h e r d e f i n i t i o n ( a n d , i n f a c t , were c a l l e d
of
using
ulthabahuLLed)
.
W e show t h a t t h e two d e f i n i t i o n s are e q u i v a l e n t .
PROPOSITION 2 . 3 8 : L e t ( E , T ) be a T V S ouek a n o n - t h i v i & y
divihion king (F, (a)
(E,T)
I 1).
valued
T h e iaLLowing c o n d i t t i o n n ahe e q u i v a l e n x .
io b a k k e l l e d .
45
TOPOLOGICAL VECTOR SPACES
16 T * in anattheti T V S t u p o L o g g i n E nuch t h a t t h e a t i i g i n hab a dundamentat b y b t e m a d r - c l a h e d n e i g h b a a hoadb, t h e n T* C T.
(b)
PROOF:
(a) * (b): L e t
V b e a -c*-neighborhood o f
a -r-closed T*-neighborhood
W1
of
0 in
such t h a t
E
W i t h o u t l o s s of g e n e r a l i t y w e may a s s u m e t h a t Choose and f i x
X
Suppose t h a t hoods o f
. ,Wn
W1,..
0 such t h a t
k =2,...,n.
hoods o f
1x1
F*, w i t h
E
Wk
By h y p o t h e s i s ,
+
C
V.
W1 i s b a l a n c e d .
a r e balanced T-closed and
Wk C Wk-l
Wk
C
neighborfor a l l
XWk-l
t h e ( b a l a n c e d ) T - c l o s e d -r*-neighbor-
-r*-neighborhood
so t h a t 'n+l
W1
< 1.
0 form a f u n d a m e n t a l s y s t e m f o r
so a b a l a n c e d r - c l o s e d
0 i n E . Choose
+
'n+l
(E,-r*)
of
'n+l
a t the origin, 0 can b e found
n'
and
By i n d u c t i o n , w e h a v e d e f i n e d a T-closed E.
i s b a r r e l l e d , U i s -c-topological.
Since (EJ)
implies t h a t (b)
+ -
string
V i s a T-neighborhood
(a): L e t
F
of
0.
C l e a r l y , F i s d i r e c t e d . By P r o p o s i t i o n 2 . 3 3 , k n o t s of a l l s t r i n g s belonging t o
F
borhoods a t
T*.
0
a r e 7-closed,
f o r a T V S topology
i t f o l l o w s from ( b ) t h a t
Now
Hence
be t h e s e t of a l l T-closed
U = (W ) i n
T*
W1
(F, 1 . 1 ) .
A string
i f every knot
Un
Let
C V
C T.
strings i n
the set
of
E.
all
B o f neighSince a l l elements i n B
form a b a s i s T*
C
T.
Hence t h e k n o t s
of 0 , i . e . o f a n y T-closed s t r i n g a r e T-neighborhoods -r-closed s t r i n g i s T - t o p o l o g i c a l , a n d ( E , T ) i s b a r r e l l e d . DEFINITION 2 . 3 9 :
n
any
( E l T) be a T V S over a valued d i v i s i o n ring
U = (U is E is c a l l e d T-bohnivohaun n n€IN a b s o r b s a l l bounded s e t s i n (E,T); t h a t i s ,
46
PROLLA
€or e a c h
n
such t h a t
IN, g i v e n a bounded s e t B whenever 1x1 6 n
E
.
B C AUn
C
E
there exists 6n>0
C l e a r l y , any r - t o p o l o g i c a l s t r i n g i n ( E l T ) is 7 - h r n i v o r o u s DEFINITION 2 . 4 0 :
(F,
I I) .
.
( E , T ) b e a T V S over a valued d i v i s i o n ring
Let
We s a y t h a t ( E l r) i s b o t n o l o g i c d i f e v e r y r - b o r n i v o r o u s
s t r i n g is r-topological. L e t ( E 1 , ~ l ) and
DEFINITION 2 . 4 1 :
( E 2 , r 2 ) b e two T V S
same v a l u e d d i v i s i o n r i n g ( F , \ - \ ) . A l i n e a r mapping is c a l l e d bounded i f
over the T:
El
+
E2
T maps bounded s e t s i n t o bounded s e t s .
By P r o p o s i t i o n 2 . 1 9 ,
every continuous
linear
mapping
is
bounded. THEOREM 2 . 4 2 :
1 e - t (E,T) be
1 1)
d i u i n i u n K i n g (F,
.
The
a T V S
ouex
a non-ZhiuLaLRq vaRued
dvlLawing ake e q u i v a l e n t :
(a)
(E,r)
(b)
e u e t y b o u n d e d L i n e a h m a p p i n g deljined u n ( E , r ) i n c a n -
i6
boxnvLogical;
tinuoun.
PROOF:
(a)
*
(b): Let
Choose a r * - t o p o l o g i c a l
Clearly, T
-1
(U)
=
because
l o g i c a l . Hence
T-’(W)
continuous a t
0,
string
(T-’(Un))
r-bornivorous,
*
T : (E,T)
+
( E * , T * ) b e a bounded
W be a b a l a n c e d r*-neighborhood o f
mapping. L e t
U = (Un)n E m i n
is a string i n
linear
0 i n (E*,T*). E * w i t h W1=W.
(E,T)
which
is
( a ), T - l ( U ) i s r - t o p o i s a T-neighborhood of 0, and T is T i s b o u n d e d . By
and b e i n g l i n e a r , T
i s continuous.
F be t h e set a l l r-bornivorous s t r i n g s in E . C l e a r l y , F i s d i r e c t e d . By P r o p o s i t i o n 2 . 3 3 , t h e s e t of all k n o t s of a l l s t r i n g s b e l o n g i n g t o F f o r m s a b a s i s 8 o f n e i g h b o r h o o d s a t 0 f o r a T V S t o p o l o g y T*. L e t B C E b e a r-bound& s e t . Then B i s a b s o r b e d by a l l e l e m e n t s i n 8 . Hence B is (b)
r*-bounded.
(a): Let
T h i s shows t h a t t h e i d e n t i t y map f r o m ( E , T )
into
(E,r*) i s bounded. By ( b ) , t h e i d e n t i t y map i s c o n t i n u o u s . T n e r e f o r e , e a c h e l e m e n t of
8 is a r-neighborhood of
0,
and soevery
47
TOPOLOGICAL VECTOR SPACES
F i s r-topological.
element i n
E v e h y m e t f i i z a b l e T V S o v e h a non-,t&ivLaePy d i V i h L o n k i n g (F,1 I ) in bo4noLvgicaL.
COROLLARY 2 . 4 3 :
PROOF:
Let
-
( E , ' I ) b e a m e t r i z a b l e T V S , and l e t
b e a bounded l i n e a r mapping. L e t t h e topology
0 in T
-1
-
= { t E E;
xn
B
=
n E IN
x E T-I(W) n
,
0 i n (E,r).
For each
n
AilUn
Hence
AnW.
€
IN,
0 and t h e r e f o r e c a n n o t b e c o n t a i n e d
and
Now t h e se-
xn $Z T - l ( W ) .
Hence
A-lT(X x) = n n
Tx
is
T(B) C
AW.
par-
and then
E W,
n
and
0
6 > 0 be such t h a t / A / 2 6 i m p l i e s T B) so t h a t / A n [ 2 6. Then T(B) C XnW. In
t i c u l a r , T(A x ) n n
Let
m.
0 i n (E*,T*) such t h a t
i s t o t a l l y bounded.
{Anxn ; n E IN
bounded. L e t Choose
+
i s bounded i n (El?). I n d e e d , hnxn
q u e n c e ( Anxn)
so
E
E defining
F* w i t h ] A , ]
W be a neighborhood of
Choose
(E*,T*)
i s a b a s i s of neighborhoods a t
i s n o t a neighborhood of
-1 ( W ) . T
E
+
d ( t , O ) < 2-"}
Then ( U n f n E T N
i s a neighborhood of
AnlUn
in
n E IN.
(E,T). L e t
(W)
An
T : (Err)
a m e t r i c on
d be
Choose a s e q u e n c e
T.
"n f o r each
vaLued
-1 a c o n t r a d i c t i o n . T h i s c o n t r a d i c t i o n shows that T (W)
must be a n e i g h b o r h o o d of
0
in
( E , T ) , and
T
i s continuous.
r b g e n e r a t e d by t h e s e t F of a l l T-closed s t r i n g s i n (E,'I) i s c a l l e d t h e hfttrong X a p o L o g y of The t o p o l o g y
REMARK 2 . 4 4 :
( E r r ) . Clearly, T T
a linear
b
T'l(V)
Let
and ( E , T ) i s b a r r e l l e d i f , a n d o n l y
C
-rb
if,
is the following: i f
mapping T : ( E J ) + (G,C) i s continuous, then b + ( G I < ) i s c o n t i n u o u s t o o . Indeed, l e t V be a ?-neigh-
b o r h o o d of
U1
,
A n o t h e r i m p o r t a n t p r o p e r t y of
= rb.
T : (E,'I )
at
b
C T
V.
0
in
Then
G.
T
Choose a < - c l o s e d s t r i n g
-1
(U)
i s a 7-closed s t r i n g i n
i s a Tb-neighborhood of B(T)
U in
G starting
E,
and
so
E
such
0 .
be t h e s e t of a l l
T V S topologies
r~ o n
48
PROLLA
that
c 11;
(1)
T
(2)
(E,q)
The s e t
F
is b a r r e l l e d .
F. Then
g e n e r a t e d by
PROOF:
l/ b e a
Let
q-closed
f o r each
T
L
-rL
-rt
Let
be t h e
for all
q
C
t
Since
E.
q t B(T)
.
Thus
is
q,
C
i
(E,q) is barrelled;
q E B ( T ) . Now
for
topology
T VS
6 B(T).
t- c l o s e d s t r i n g i n
is q - t o p o l o g i c a l f o r e a c h
U
E which are n - t o p o l o g i c a l
of a l l s t r i n g s i n
q E B ( T ), is directed.
every
therefore
F , and so
U E
U
is T t - t o p o l o g i c a l .
PROOF: L e t
V
€,-topological
6
b e any string
for all
C q
6-neighborhood o f U = (Un)
q E B(T:.
of
Tt-neighborhood DEFINITION 2 . 4 5 :
0
-
,
Hence
The t o p o l o g y
b a f i t e L L e d t o p o L v g y of
with
U1
C
and,
U1
in
0
a fortiori,
the
V
PROOF:
t
T : (E,T )
v +
be t h e f i n e s t T V S topology
(G,v)
T : ( E , T ~ )+ ( G , v b )
i s continuous.
T : (E,T
5
t
C
v
t
+
(GI()
and so
By ( a ) ,
on (T
t b
v = v
b
,
i.e.
is continuous, 5
T : (E,-rt)
+
(G,C
t
)
C
v
is
v, i . e .
=
)
vb
T
t
.
C v.
barrelled.
v
E
Lineafi
such t h a t
G
i s c o n t i n u o u s , and t h e r e f o r e
this i t f o l l o w s t h a t
a
T.
L e t T : ( E , T ) + ( G I < ) be. a c o n t i n u o u b T : ( E , Tt ) * ( G , E t ) i n c o n t i n u o u n .
Let
is
ac)i)OC,&Ltted
PROPOSITION 2 . 4 6 :
map. T h e n
a
Then U E F, b e c a u s e
V.
is called
T~
Choose
E.
B(C).
is continuous too.
Hence From
Since Hence
49
TOPOLOGICAL VECTOR SPACES DEFINITION 2 . 4 7 :
Clearly,
T'.
T C T'.
t h e same bounded s e t s . I n f a c t ,
Moreover
and
T
generates have
'T
(E,-ra) is a bornological space,
E with t h i s property.
-ra
i s t h e c o a r s e s t b o r n o l o g i c a l t o p o l o g y on
and
E which i s
finer
The s p a c e ( E r r R ) i s c a l l e d t h e a b d o c i a t e d b o f i n o L o g i c a L
T.
Apace o f ( E , T ) . A T VS ( E , T ) i s b o r n o l o g i c a l i f , R T = T .
If
t h e s e t of a l l
is t h e f i n e s t T V S topology
T'
on
than
*I),
i s d i r e c t e d . Hence i t
T-bornivorous s t r i n g s i n ( E J ) a T V S topology
(F,[
( E , T ) i s a TVS over
If
T : (E,T)
T : ( E , Ta )
+
(G,€,)i s
+
a
a continuous
and
only i f ,
l i n e a r mapping, t h e n
( G , < ) i s c o n t i n u o u s too.
PROOF: S i n c e ( C r u x ) i s b a r r e l l e d , i t s u f f i c e s t o show
that
has a fundamental system of p*-closed neighborhoods o f
i s t h e metric d e f i n i n g
p , choose
Un
p
0. I f d
so t h a t
i s a f u n d a m e n t a l s y s t e m of u - c l o s e d b a l a n c e d n e i g h b o r h o o d s of 0 i n (G,p),
s a t i s f y i n g (a) through
For each
n E IN,
p* is b a r r e l l e d ,
x1 E Un+l
with
j
Since ( E x i ) . 3 i =1 y E G.
is p-closed,
x
-
x
E
Tn+2
xj
E
Un+j
j i =1
y E Un
xi
belongs t o
.
x = y.
Indeed, s i n c e
of
Since
Un.
i s p*-topological.
.
+ Tn+,)
=
By i n d u c t i o n , j and x xi i=l
'
i s a p-Cauchy s e q u e n c e ,
Since
W e claim t h a t
~
Then, b y t h e a b o v e r e m a r k (x
d e f i n e a sequence ( x . ) w i t h 7
some
be t h e u*-closure
Tn
the s t r i n g (Tn)n
L e t x E Tn+l. Choose
Let
Ic) o f Theorem 2 . 1 5 .
j
2
i =1
it Un+i
p* C p r
w e can Tn+j+l.
converges C Un
j
( Z
$3.
and
to "n
x i ) converges
i =1
50
to
PROLLA
y i n (G,u*).
x - y
Hence
j EJN.
Now
j
-
x Tk
E
for a l l
x - y # 0. Since
Assume
belongs t o
xi i=l
2
Tn+j+l
for
all
k E IN.
i s a H a u s d o r f f t o p o l o g y , there
p*
i s some p * - n e i g h b o r h o o d V of 0 i n G s u c h t h a t ( x - y + V) n V = @. S i n c e p * C p , t h e r e i s some k t IN s u c h t h a t ( x - y + V ) n U = @. Therefore
x
-
y
9
,
Tk
k
a contradiction.
T h i s e n d s t h e p r o o f of Lemma 2 . 4 8 . L e t (F,1 . 1 )
b e a n o n - t r i v i a l l y v a l u e d d i v i s i o n r i n g . A con-
t i n u o u s l i n e a r mapping Hausdorff
set
TVS
(G,ii)
{(x,Tx); x E El
T from a T V S
over (F,1 . i s closed
p r o d u c t t o p o l o g y . The s o - c a l l e d conditions
1)
i n t h e space T: E
i s c o n t i n u o u s from ( E J )
on
11"
with
with closed
G
-+
the
states
are a s a b o v e ) . N o t i c e
d o r f f T V S ( G , u ) h a s c l o s e d graph i f , and o n l y i f
a Hausdorff T V S topology
i.e. the
from a T V S ( E , T ) i n t o a Haus-
G
-+
E x G
into a
.theotem
cLobed ghaph
u n d e r w h i c h a l i n e a r mapping T: E
1)
h a s a c f o h ~ dg t t a p h ,
g r a p h i s c o n t i n u o u s ( w h e r e (E,T) a n d ( G , p ) t h a t a l i n e a r mapping
o v e r (F,1 .
(EJ)
G such t h a t
,
there exists
p* C
and T
!J
i n t o (G,p*),
L e t u s now p r o v e a " c l o s e d g r a p h t h e o r e m " .
PROOF:
By t h e r e m a r k s p r e c e d i n g t h e s t a t e m e n t
t h e r e e x i s t s a Hausdorff T V S topology
p*
of
on
Theorem 2 . 4 9 , G
such
i ~ *C and T i s c o n t i n u o u s from ( E , T ) into (G,p*). c o n s i d e r t h e a s s o c i a t e d b a r r e l l e d t o p o l o g y ( p * I t = po
.
i s c o n t i n u o u s from ( E , T ) i n t o ( G r p o ) Now, w e have
because
po C
u,
with
is Hausdorff
DEFINITION 2 . 5 0 :
Let
(F,
.
1. I )
r i n g a n d l e t ( E , T ) and (G,v)
po
(Recall th at
T
that
Let us Then T
is barrelled).
b a r r e l l e d . By Lemma 2 . 4 8 ,
po=ii,
be a n o n - t r i v i a l l y valued d i v i s i o n be t w o
TVS
over
i t . Then L ( E ; G )
51
TOPOLOGICAL VECTOR SPACES
denotes into
t h e v e c t o r s p a c e of a l l c o n t i n u o u s l i n e a r
maps
of
G.
DEFINITION 2.51:
G-topologies) : L e t
(
b e a f a m i l y of bound-
ed s u b s e t s o f ( E , T ) c l o s e d u n d e r f i n i t e u n i o n s , a n d l e t a fundamental system of neighborhoods of
6
S E
and
i n (G,v).
0
B
be each
For
let
V E B
Clearly, the set
o f a l l such
8~
= C ( E ; G ) . L e t u s assume
2.15.
E
B
W(S,V) i s a f i l t e r b a s i s o n
s a t i s f i e s (a) t h r o u g h
(c) of
Theorem
Then
# 0,
when U + U C V. S i n c e W(S,AV) = AW(S,V) f o r a l l A s a t i s f i e s ( b ) , and e a c h W(S,hV) i s b a l a n c e d . Let and
W(S,V) b e g i v e n . By P r o p o s i t i o n 2 . 1 9
(G,v)
,
and s o t h e r e e x i s t s
f ( S ) C XV,
i.e.
6 > 0
f ( S ) i s bounded
1X1 2
such t h a t
f E W(S,XV) = A W ( S , V ) .
BG
f E C (E;G)
Hence
6
W(S,V)
in
implies
is
ab-
sorbing. W e h a v e shown t h a t t h e f i l t e r b a s i s
t i e s ( a ) t h r o u g h ( c ) of Theorem 2 . 1 5 , topology over
(F, I
over
TG
1)
L(E;G)
f o r which
86
such t h a t
BG
s a t i s f i e s proper-
and s o t h e r e i s a ( u n i q u e ) ( ~ : ( E ; G ) , T c)
is a
TVS
i s a fundamental system of
neigh-
borhoods. The m o s t i m p o r t a n t e x a m p l e s o f
G-topologies a r e the
fol-
lowing :
i s t h e s e t of a l l l j i n i t e s u b s e t s o f
(a) case (b)
G
TG
i s c a l l e d t h e ,ttopoLogy
E.
In this
0 6 hhnpLe convehgence.
i s t h e f a m i l y of a l l t o t a l l y bounded s u b s e t s
of
i s t h e f a m i l y of a l l compact s u b s e t s o f
In
E.
(c)
G
t h i s case
~6
CUnVehgenCe.
is c a l l e d t h e Ropoeogy
06
E.
compac-t
PROLLA
52
i s t h e f a m i l y of a l l b o u n d e d s e t s of E . I n t h i s case T G i s c a l l e d t h e k o p o e o g y 0 6 unidohm ~ o n v c 4 -
(d)
and w e
g e n c e on bounded b e t b o r t h e b t h O M g t o p o e o g y , w r i t e 1: b ( E ; G ) t o d e n o t e ( & ( E ; G ) , -iG 1 . DEFINITION 2.52: if
A subset
i s s a i d t o be
H C 6(E;G)
i s b o u n d e d i n t h e s p a c e (6( E ; G ) ,
H
In particular, i f
)
G-bounded
.
converH i s p o i n t w i n e bounded or h i m p L y bounded i f
gence, w e say t h a t
it is
T~
is t h e topology of simple
T~
-bounded.
P R O P O S I T I O N 2.53:
LeL
Then k h e doRRu#ing ahe equhm-
H C 6(E;G).
Len.t:
in
(a)
H
(b)
e a c h neighbo4hood V o d abnohbcr e v e h y S E 6 .
(c)
Foh each
6-bounded.
Fa4
,
S E
0 i n (G,v),n If-'(V),
{f(S); f E
U
HI
f
E
bounded
ib
HI i n
(G,v).
PROOF: ( a ) * ( b ) : S i n c e w e h a v e a s s u m e d t h a t is
6 > 0 such t h a t
hV.
Hence (b)
all
* (c):
If
Hence
f t H.
(c) * (a): L e t
1x1 1.
that
1x1
>
6
S C A n {f-'(V);
6
I*1)
implies
H C AW(S,V),
non-
f(S) C
i.e.
f
€
HI
f E C
HI,
then
f(S)
f ( S ) C AV
C
AV f o r
hV. 8 > 0
W(S,V) be given. There e x i s t s
implies
is
f E H}.
S C A n {f-'(V); U{f(S);
(F,
V i s b a l a n c e d . By (a) , t h e r e
t r i v i a l l y v a l u e d w e may a s s u m e t h a t
for all
f
E
H.
such
Hence
H C AW(S,V).
PROPOSITION 2.54:
Let
H C d: ( E ; G ) .
Then t h e doLeowing me eguiva-
RenL: (a)
H
(b)
Foh each neighbohhood
i b eyuicantinuoub.
v 06
0 i n (G,v), n
If-'
(V); f E
HI
53
TOPOLOGICAL VECTOR SPACES
06
in a neighbohhood
0 i n (E,T).
F o h e a c h neighboxhood V u d 0 i n ( G , v ) thehe exints a neighboxhood u 06 0 i n ( E , T ) n u c h t h a t
(c)
PROOF: C l e a r . COROLLARY 2 . 5 5 :
ed d o h e u e h y
An e q u i c a n t i n u o u n A u b h e t o d
d: ( E ; G )
i d
(%bound-
G-topology.
PROPOSITION 2 . 5 6 :
L e t H C E ( E ; G ) b e an e q u i c o n t i n u o u n n u b n e t . The h e n t h i c t i o n n t o H o d t h e doLLowing t o p o l v g i e n a t e Rhe name: (a)
t h e t o p o l o g y o d nimpLe COnUehgence;
(b)
t h e t o p o L o g y ad unidohm c o n u e h g e n c e on t o t u L R y bounded nubnetn.
PROOF:
Let
fo
t a l l y bounded, and to
and
E H,
W(S,V) b e g i v e n , w h e r e
V i s a n e i g h b o r h o o d of
R . Choose a n o t h e r o n e
to-
0 i n ( G , v ) , belonging
such t h a t
R
W E
is
S C E
W + W + W C V ,
a n d , by e q u i c o n t i n u i t y of i n (E,?) s u c h t h a t
f(U)
t a l l y bounded, t h e r e i s S C So
L e t now
with
+
U.
Hence
C
a symmetric neighborhood
U of
for a l l
is
f
a finite
S
E
H.
Since
S
such
subset
0 to-
that
f E W(So,W) i m p l i e s
g E H n [fo
U
W
C
So
+
W ( S o f W ) ] . Then
f E W(So,W). S i n c e
because
H,
i s Symmetric.
g E H and f o r w e see t h a t
f = g
-
Thus
f(S)
C
W
+ W +
W
C
g
V,
= f
0
+
f,
i. e .
54
PROLLA
f E W(S,V).
g belongs to
Therefore
fo
+
W ( S , V ) , and
i f o + w(So,w)l c f o + w ( s , v ) .
H n
L e t (F,1
- 1)
be a non-XhiuiaLlq v a l u e d d i v i s i o n k i n g be. LWu T V S o u e h i ; C u k t h ( G , v ) Haudolr6zj. €1 C X ( E ; G ) i n e q u i c o n t i n u o u b and H1 i 0 t h e c e o s u h e 06 H i n GE (.in t h e ptroduct t o p a ~ u g g ) ,t h e n HI C d:(E;G) and H1 i n eyuicontinuuuh.
LEMMA 2 . 5 7 :
a n d &ex ( E , r ) and
PROOF:
(G,v)
€ t H1, t h e r e e x i s t s a n e t
If
i n t h e product topology.
+
pfa(y)
+
i . l f ( y ) = f (Ax
Af (x)
+
+
n e i g h b o r h o o d of
+
pf (y)
I t follows t h a t
.
in
fa(Ax
is linear. L e t
f
+
liy) = A f a ( x ) +
V be a
+
Xf(x)
v-closed
H i s e q u i c o n t i n u o u s , there exists
Since
G.
H such t h a t fa + f
Since ( G , v ) i s Hausdorff,
py). Hence 0
fa i n
a r-neighborhood U of t h e o r i g i n i n E s u c h t h a t g ( U ) C V f o r a l l g E H . L e t now x E U and f E H1. T h e r e e x i s t s a n e t fa in +
H such t h a t
f ( x ) , and
Therefore tinuous; C
fa
f
+
fa(x) t V
f(U)
C V
i n t h e p r o d u c t t o p o l o g y . Hence f a (x)
f o r all
i n p a r t i c u l a r each
f E HI
f
HI
L(E;G).
L e t (F,1 .
THEOREM 2 . 5 8 :
1)
+
f ( x ) E V, because V i s closed.
implies
and s o
is e q u i c o n is continuous, i . e . H1
H1
be. a non-t&iviaLEy vaeued d i v h i o n 'ting
avid &eA ( E , r i ) and ( G , v ) be bwu T V S o v e & it. 7 6 ( E , q ) i s bakheLLed, t h e n e a c h p o i n X w i s e b o u n d e d s u b s e t H u 6 L(E;G) i s eyuicontinuuu4. PROOF:
V
Let
be a fundamental s y s t e m o f v-closed neighborhoods
of t h e o r i g i n i n For each through
V E V,
V, W
G s a t i s f y i n g ( a ) t h r o u g h (c) o f Theorem let
w
= n {f-'(V);
f E H}.
r u n s through a f i l t e r base
Then, a s
6 in
2.15. runs
V
E which can
be
taken a s a fundamental system of 0-neighborhoods f o r a T V S topology
5 o n E l t h e b o u n d e d n e s s of
H ensuring t h a t each
W
is
a b s o r b i n g . By t h e c o n t i n u i t y o f e a c h Since ( E , q )
f E H, W is Q-closed. i s b a r r e l l e d , by P r o p o s i t i o n 2 . 3 8 w e h a v e 5 C q.
Hence W i s a n e i g h b o r h o o d o f by P r o p o s i t i o n 2 . 5 4 .
0
i n ( E , q ) , and H i s equicontinuous,
55
TOPOLOGICAL VECTOR SPACES
PROOF:
Let
o u s . Now
H =
{fa;
c1
E A}.
f belongs t o t h e c l o s u r e of
t o p o l o g y ) . By Lemma 2 . 5 7 , fa
+
f
By Theorem 2 . 5 8 ,
f
€
d: ( E ; G )
H
in
H i s equicontinu-
GE
( i n the product
, and by P r o p o s i t i o n
u n i f o r m l y on e v e r y t o t a l l y bounded s u b s e t o f
I n I y a h e n [ 34 ]
,
2.56,
(E,rl).
u l t r a b o r n o l o g i c a l (and q u a s i - u l t r a b a r r e l l e d )
spaces w e r e introduced. Following [ l 1
w e dropped
the
prefix
u l t r a . I y a h e n ' s d e f i n i t i o n of an u l t r a b o r n o l o g i c a l space i s t h e
T V S (E,T) is called ultrabornological
following: a
bounded l i n e a r map f r o m ( E J ) [ 3 4 ], p.
D e f i n i t i o n 4.1,
if
every
(see
i n t o any T V S i s c o n t i n u o u s
2 9 8 ) . Our Theorem 2 . 4 2
shows t h a t E f i -
n i t i o n 2.40 a n d I y a h e n I s d e f i n i t i o n a r e e q u i v a l e n t . The o t h e r c l a s s o f T VS i n t r o d u c e d by I y a h e n
is
s p a c e s : t h o s e T V S i n which e v e r y
quasi-ultrabarrelled
that
of
borni-
i s a n e i g h b o r h o o d of t h e o r i g i n . This suggests
vorous u l t r a b a r r e l the following. DEFINITION 2 . 6 0 :
(F,1 .
I).
( E , T ) be a T V S o v e r a v a l u e d d i v i s i o n ring
Let
W e s a y t h a t ( E l ? ) i s quani-ba44eLRed i f e v e r y - r - c l o s e d
7-bornivorous s t r i n g i n
E
i s T-topological.
C l e a r l y , e v e r y b a r r e l l e d and e v e r y b o r n o l o q i c a l
space
is
quasi-barrelled. If
(F,1
(El?)
- 1) ,
i s a T V S o v e r a n o n - t r i v i a l l y valued d i v i s i o n ring
t h e s e t of a l l s t r i n g s w h i c h a r e
T
C . learly,
~
.ra c
Ta ;
Since
T C
-ra.
because
T'
same bounded s e t s a s and only i f ,
T
= T
a
both
-r-closed
Hence i t g e n e r a t e s a T VS
T-bornivorous i s d i r e c t e d . T
and
'T
and
topology
h a v e t h e same bounded sets,
i s t h e f i n e s t T V S t o p o l o g y on E w i t h t h e
.
T.
The s p a c e ( E , T ) i s q u a s i - b a r r e l l e d i f ,
A s i n t h e c a s e of b a r r e l l e d n e s s , o n e c a n d e f i n e a n u n u c i a t e d
56
PROLLA
qiinhi-batrtleLLct1 t a p a e a g q
-ryt
it is the
3 T:
c o a r s e s t quasi-
b a r r e l l e d topology which i s f i n e r t h a t T. To c o n s t r u c t proceeds a s f o l l o w s : gies
on
rl
one
topolo-
c n;
T
(2)
(E,n)
is a q u a s i - b a r r e l l e d .
F of a l l s t r i n g s i n
The s e t
E which a r e n - t o p o l o g i c a l f o r
i s d i r e c t e d . L e t -iYt F . ‘Then T~~ C n f o r a l l
be t h e
Q t QB(-r)
g e n e r a t e d by
PROOF:
.rqt
E such t h a t
(1)
every
Q B ( - r ) b e t h e s e t of a l l T V S
let
U be a s t r i n g i n
Let
Since
topology
T VS
Q E QB(-r).
E which i s c l o s e d and b o r n i v o r o u s
w i t h r e s p e c t to
7qt.
with respe ct to
17, f o r e v e r y
-tqt
C r(
b a r r e l l e d , U is q-topological.
n,
U i s c l o s e d and b o r n i v o r o u s
Since ( E , q ) is quasi-
E QB(T).
Hence
F,
and t h e r e f o r e
U is
t u p u l o g y withi E C 17 5 c T q t . I n p a h t i c u t a h , 7 c Tqt.
aft
U
E
Tqt-topological.
Let
be any <-neighborhood of
V
5-topological
5
cause
C
PROOF: L e t that
c
Tqt
c
n of
,
a
beU,
T’.
Um C V, f o r some
m
E Dl.
the s t r i n g
There i s
E.
II E F , and so T~
-rqt.
Since
T
C
and
17
U is n-topological,
T h i s proves t h a t
C
in
0
closed and 7 - b o r n i v o r o u s ,
r- t Q B ( T ) . Hence
is
(E,q)
for
( E , 7 P ) i s q u a s i - b a r r e l l e d and t h e r e f o r e
Hence
T~~
a
some
and such
Urn i s a rqt-neighborhood
Since ( E , 7 ’ )
space,
C
Choose
0.
U = (Un), which i s
quasi-barrelled,
E.
with
be a Ta-neighborhood of
V
in
0
U1 C V . Then U E F , t Q B ( 7 ) . Hence ’ and f o r t i o r i
11 = ( U n )
r), for a l l
Ta
( C )
string
string
neighborhood
is a
T VS
i-h any
E ~ ~ ( 7 tlzen 1 ,
r)
PROOF:
E
16
(b)
is every of 0 .
bornological T’
E
QB(T).
T’.
The analogue of Theorem 2 . 4 9 i s
true
for
quasi-barrelled
TOPOLOGICAL VECTOR SPACES
57
spaces :
L e t ( E , T ) b e a quani-bahhetted T VS oveh a n o n , t h i w i a e l y w a t u e d d . i v i 4 . i o n h i n g ( F , I I ) T h e n e v e h y bounded f i n e a h t n a p p i n g T 06 E i n t o a compteLe t n e t h i z a b & e T V S ( G , w ) oveh ( F , I I ) , W i t h d o b e d ghaph in c v n . t i n u o u b . THEOREM 2 . 6 1 :
.
-
Theorem 2 . 6 1 i s a c o n s e q u e n c e o f t h e f o l l o w i n g lemmas.
LEMMA 2.62: A b o u n d e d l i n e a h mapping derjined o n a q i i a n i - b m & e d npace. in Meahley con.tiiquouh. LEMMA 2 . 6 3 :
A L i n e a h mapping d e d i n e d o n any T V S
and ~ i L hv d u U
i n a campCete m e , t k i z a b l e T VS, w h i c h i n n e a h t y C o n t i n u o u c S
cloned ghaph,
hab
i 4
and
continuoun.
B e f o r e p r o v i n g t h e a b o v e lemmas l e t u s r e c a l l t h a t a l i n e a r mapping
T : (E,r)+ ( G I < ) i s
<-neighborhood in
E
V
nea4Ly con-tinuuub i f ,
of t h e o r i g i n i n
i s a T-neighborhood
PROOF O F LEMMA 2 . 6 2 :
Let
for
t h e r - c l o s u r e of
GI
c-neighborhood of 0 i n such t h a t
T : (E,7)
G.
+
( G I < ) b e a bounded
For each
T i s bounded, t h e s t r i n g
let
n E IN
Wn
linear V
Choose a 6 - c l o s e d s t r i n g
U1 C V . S i n c e
i s r-bornivorous.
(V)
0.
of
mapping d e f i n e d on a q u a s i - b a r r e l l e d s p a c e ( E , r ) . L e t G
T
every -1
be a
U = (U ) in n (T-l(Un))
b e t h e 7 - c l o s u r e of
-1 ( U n ) .
Then W = (W ) i s a r - c l o s e d 7 - b o r n i v o r o u s s t r i n g i n E. n S i n c e ( E J ) i s q u a s i - b a r r e l l e d , W i s 7 - t o p o l o g i c a l a n d W1 i s a T
T-neighborhood T
-1
of
0.
Since
( V ) i s a r - n e i g h b o r h o o d of
T
-1
(U,)
0 in
T-'(V)
,
E l and
T
C
t h e r - c l o s u r e of
is nearly
con-
tinuous. B e f o r e p r o v i n g Lemma 2 . 6 3 l e t
us introduce the
following
definition: DGFINITION 2 . 6 4 :
A TVS
(GI<)
i s c a l l e d Br-compLete
n e a r l y c o n t i n u o u s l i n e a r mapping
if
every
T I w i t h c l o s e d g r a p h , from a n
a r b i t r a r y T V S ( E , r ) i n t o (G,S;) i s c o n t i n u o u s .
58
PROLLA
Using t h e above d e f i n i t i o n ,
as f o l l o w s :
t h e s t a t e m e n t of Lemma 2.63 reads
cwek~yc o m p t e t c m e t h i z u b e e T V S
Since every complete metrizable
,
Theorem 2 . 3 7 )
,Lid
is
T VS
B
-cvmpLete. r (see
barrelled
Lemma 2 . 6 3 f o l l o w s f r o m Lcmmas2.65 and 2.66
below.
A H a u s d o r f f T V S ( G , E ) i s c a l l e d an indhu-sApace i f , f o r e v e r y coarser I I a u s d o r f f T VS t o p o l o g y 1-1 o n G w e t t have p 3 5 o r , e q u i v a l e n t l y , p t = F; DEFINITION 2.67:
.
PROOF O F LEMMA 2 . 6 5 : let
i~
L e t ( G , C ) be a c o m p l e t e m e t r i z a b l e T VSand
b e a H a u s d o r f f T V S t o p o l o g y on
t h e i d e n t i t y mapping I : ( G , p )
+
h a s a c l o s e d g r a p h a s a m a p p i n g from ( G , p t )
' 1-I.
i-lt
5
C
pt.
By Theorem 2 . 4 9 ,
t h e mapping
By D e f i n i t i o n 2 . 6 7 ,
PROOF OF LEMMA 2 . 6 6 :
infra-s-space. -T
such t h a t
T~
Let C
T.
Let T C
is
I
(7; d e n o t e s t h e
0
0.)
of
(G,C)
,
continuous,
a
be
Hausdorff
since i.e.,
barrelled
b e a H a u s d o r f f T VS t o p o l o g y T V S t o p o l o g y on
as fundamental system of 0-neighborhoods T -neighborhoods
onto
( G I < ) i s an infra-s-space.
(E,-ro)
-r0
It s t i l l
has c l o s e d graph.
(G,C)
5 . Then
coarser t h a n
GI
Since
T~
the
on
E
E w h i c h has
-r-closures of t h e
is barrelled,
-T
i s also
T~
b a r r e l l e d . IIence
( E , T ~ ) is an infra-s-space
On t h e o t h e r h a n d ,
Hence T
= -c0
-T t (T ) = 0
.
T'
0
= T
0
.
Now w e h a v e
-T
0
=
TT 0
--T
and
C T C T~
T
I
C
T
~
i.e.
I t remains t o prove t h e following.
LEMMA 2 . 6 8 :
Let (E,ro) b e a Haundohdd
TVS
duch t h u t , doh
unq
.
59
TOPOLOGICAL VECTOR SPACES
coakbeh Haubdohdd TVS Lopo&ogy have
= T
T
. 0
Then ( E , T ~ )
on
T
b u c h ,that
E
7;
C T,
we
Br-complete.
i d
PROOF: S u p p o s e ( E , - c o ) i s n o t B r - c o m p l e t e . c o n t i n u o u s l i n e a r mapping T : ( G , p )
-+
There e x i s t s a n e a r l y
( E , T ~ )w , ith closed graph,
which i s n o t c o n t i n u o u s . Hence t h e f i n e s t T V S t o p o l o g y T on E , such t h a t T : ( G , p ) gy and
$
T
T
( E r r ) i s c o n t i n u o u s , i s a Hausdorff topolo-
--f
y:
W . e claim t h a t
~
C T.
of 0 i n
E.
Since
i s nearly continuous, T
0.
T
Choose a -ro-neighborhood
BY c o n t i n u i t y o f Thus
mapping t i o n of
T
-1
T-~(u) c
T,
is a
(V)
-1
?J
L e t V b e a "s-neighborhood 0 U of
such t h a t
0
i s a v-neighborhood
(U)
T,
-T
T~
T.
C
i s B -complete,
p-neighborhood of
Thus
T
QED.
r
0
= 7 :
C
V.
of
T-'(V). t h e o r i g i n , and
i s c o n t i n u o u s f r o m ( G , ? J ) i n t o (E,?:).
T
zT
the
By t h e d e f i n i -
a c o n t r a d i c t i o n . Hence ( E , ' r 0 )
The r e s u l t s c o n t a i n e d i n Theorem 2 . 4 9 and 2 . 6 1 a r e known a s the
cLobed
theohem
gtaph
for barrelled
and q u a s i - b a r r e l l e d
e s p a c e s . L e t u s now s t u d y t h e s o - c a l l e d o p e n m a p p i n g t h e o k e m . W
s t a r t with the following d e f i n i t i o n . DEFINITION 2 . 6 9 :
L e t ( E , T ) and
( G , u ) b e two
T V S o v e r t h e same
G v a l u e d d i v i s i o n r i n g (F,I ] ) . A l i n e a r m p T : E i s s a i d t o b e n e a k l y ( o r u l m v b , t ) upen i f for e a c h T - n e i g h b o r h o o d
non-trivially
U
of
0 in
0 in
+
E , t h e u - c l o s u r e of
T(U) is a
1.I-neighborhood
of
G.
PROPOSITION 2 . 7 0 :
(a)
Any lineah m a p p i n g u n t u a bahacLLed Apace
i b neahly open. (b)
Lei
T :E
oh necond catcgohy i n PROOF:
(a)
Let
a T-topological G,
v
=
( T ( U n ) 1.I )
G.
T : (EJ)
b a r r e l l e d space. L e t
be a l i n e a h mapping buch t h a t
G
-+
Then
--f
T i n neatly open.
(G,p) be a
l i n e a r mapping o n t o
U b e a T-neighborhood of
string
U = (U,)
i s a u-closed
T(E) i n
with
U1 C U .
string in
G.
0 in Since
Since
E.
a
Choose
T is onto (G,~.I)
is
60
PROLLA
V
barrelled, (b)
> 1. S i n c e
T i s n e a r l y open.
Hence
U b e a T-neighborhood
Let
1x1
with
i s p-topological.
of
0 in
E.
X
Choose
E
F
i s absorbing, w e have
U
u xku .
E =
l)k Therefore
T ( E ) i s of s e c o n d c a t e g o r y i n
Since
b o r h o o d of
f o r some k .
0
p-neighborhood
of
0 in
By C o r o l l a r y
T(U,) =
n
y
E
T(Un)
for a l l
b a l a n c e d neighborhood o f for a l l
xn
n
0 , so
--t
Hence
L
1. Choose
T(xn)
0 in
xn E Un
0, b e c a u s e
- y E V, because
y E V. T h i s shows t h a t
Xk-T(U) 2.3,
is a
T(U)
p-neigh-
is
a
G.
n) 1
PROOF: L e t
GI
n
{OI.
1. I,
and l e t V b e a c l o s e d and
(G,p). Then so t h a t
( y + V) n T ( U n ) # @ T ( x n ) - y E V. Now
T i s continuous.
V i s closed.
Since
V
i s balanced,
y = 0 , a s ( G r p ) i s a Hausdorff
T VS.
W e a r e now r e a d y t o p r o v e t h e Open Mapping Theorem.
b e a R i n c a k , c o n t i n u o u s , and n e u h l g o p e n m a p p i n g Bkom a cumpCEte m e t h i z a b L e T V S (E,T) into a Huu~duh6d T V S ( G r u ) . T h e n T i b o p e n , i . e . , T maps open n e t b i n t o open THEOREM 2 . 7 2 :
bCk6.
1tL
T
61
TOPOLOG I CAL VECTOR SPACES
PROOF:
I t i s s u f f i c i e n t t o show t h a t f o r some f u n d a m e n t a l system
8 of ?-neighborhoods of
a neighborhood of
in
0
El U
0 i n (G,u). C h o o s e
d e c r e a s i n g sequence and e a c h o p e n , Wn
.
L e t then
.
Wk+2
Un
implies that
B
B so t h a t
R
i s closed. S i n c e
i s a n e i g h b o r h o o d of 0 i n ( G , u ) ,
= T(Un)
If w e show t h a t
complete
E
T(Un) 3 W n + l ,
for all
.
x1
y E Wk+l
Choose
1,
n
=
is
T(U)
is a
(U,)
T is nearly
for a l l n L 1 . is
t h e proof
s u c h t h a t y-T(xl) E
'k+l
By i n d u c t i o n , we c a n d e f i n e a s e q u e n c e (x. ) with x , E 3 J
s+j
and
m
W e c l a i m t h a t t h e p a r t i a l sums o f t h e s e r i e s
form a
Cauchy s e q u e n c e i n
of 0 i n
E.
There e x i s t s
Now,
for all
'=O
p
n
2
E. 0
IN s u c h t h a t f o r a l l
E
'=O
P Xn+i
i
'k+n+i
'k+n T h i s p r o v e s o u r claim. x E E
n > n 0'
0
P i
E xi i =1 I n d e e d , l e t V be a T - n e i g h b o r h o o d
+
'k+n
Since
'k+n
c
v.
'
P
+
'k+n-l (E,T)
'k+n-l
"k+n+i i=l
c v.
i s complete, t h e r e e x i s t s
such t h a t m
x = On t h e o t h e r h a n d ,
Since and
n
Uk
f o r each
2 xi. i=l p 2 0 , w e have seen t h a t
i s c l o s e d , t h i s shows t h a t
1,
x E Uk.
NOW,
for all p 2 0
62
PROLLA
n+w
Letting
p
+
m,
we get Y -
for a l l
n
2
for a l l
n
2
1. By Lemma 2 . 7 1 ,
C
T(Uk) f o r a l l
i.e.
,
Wk+l
THEOREM 2.73:
1. S i n c e
Lct
p e e t e . meLttrizable T Thevi
is continuous, t h i s implies t h a t
T
k
y = T ( x ) . But
1. 1,
T(x) E
T(Uk);
QED.
T be. a c a n . t i n u o u A L i n e a h m a p p i n g 6hom a cmmV S ( E , T ) o n t o ci 6 a t t e l L e d Haudohdd T V S ( G , p ) .
T Xd o p e n .
PROOF:
By p a r t ( a ) o f P r o p o s i t i o n 2 . 7 0 ,
COROLLARY 2.74: T VS.
PROOF:
Let ( E , T )
and
T
i s n e a r l y open.
(G,p) be. Lwu c o m p L e t e rnethizab&e E anto C open.
T h e n any continuous L i n e a t m a p p i n g gtlum
By Theorem 2 . 3 7 ,
(G,p)
is barrelled.
L e t T b e a c o n . t i n u o u d L i n e a h m a p p i n g 6tom a comp l e t e . mctfi-izabLe T V s ( E , T ) o n t o a B a i t e Haundofidd T V S ( G , p . ) . Then T i.) o p e n .
COROLLARY 2 . 7 5 :
PROOF:
By Theorem 2 . 3 7 ,
( G , L J ) i s b a r r e l l e d . O r e l s e , by P r o p o s i -
of s e c o n d c a t e g o r y i n i t s e l f , T i s n e a r l y o p e n and t h e n a p p l y Theorem 2.72. t i o n 2.70,
( b ) , and t h e f a c t t h a t
a Baire s p a c e
COROLLARY 2.76: p
is
L e t (E,T) be. a c o m p L e t e methizablc T V S and Let be any tiausdofi66 T V S Z o p o L o g y on E, w i L h 1-1 C T a n d b u c h
t h a t ( E , p ) i n 4a.kfitLEe.d.
Then
p = 7.
63
TOPOLOGICAL VECTOR SPACES C l e a r l y , t h e C o r o l l a r y a b o v e i s j u s t Lemma
REMARK 2 . 7 7 :
( t h e C l o s e d Graph Theorem)
T h i s shows t h a t Theorem 2 . 4 9
2.48.
a
is
C o r o l l a r y o f t h e Open Mapping Theorem.
L e t T and p be ti^ c o m p l e t e m e t f i i z a b L e T V S t o p v l o g i e n on a vec,toh n p a c e E. 7 A p C T , t h e n = T.
COROLLARY 2 . 7 8 :
L e t ( E , T ) b e a c o m p l e t e me.tkizabLc T VS and L e t b e a Haundok~A T V S . L e t T be a c o n t i n u o u n L i n e a h mapping E i n t o G n u c h t h a t T E ) i n 06 ~ e c o n dcattegohy hn G. T h e n i . n open and o n t o .
THEOREM 2 . 7 9 : (G,p)
06 T
PROOF: By P r o p o s i t i o n 2 . 7 0
i s o p e n by Theorem 2 . 7 2 .
( b ) , T i s almost o p e n .
Hence
Therefore
T
T ( E ) i s o p e n i n G, and t h e r e f o r e
a b s o r b i n g . I t nows f o l l o w s t h a t T i s o n t o , s i n c e T ( E ) i s i n v a r i a n t under scalar m u l t i p l i c a t i o n .
L e t ( E J ) be a c o m p l e t e m e t h h z a b L e T V S and P e t ( G , p ) bc a Haundokdlj T V S . F o k a n y T E L ( E , G ) , e i t h e t T(E) i d ul; Rhe A i h h t c a z e g o h y in G o h T ( E ) = G .
COROLLARY 2.80:
Theorem 2.58 i s t h e e s s e n t i a l i n g r e d i e n t i n t h e p r o o f of the Banach-Steinhaus
Theorem ( 2 . 5 9 ) .
For spaces o v e r
IR
or
c,
W a e l b r o e c k [ 9 5 ] t a k e s i t a s a d e f i n i t i o n o f b a r r e l l e d s p a c e s (of course he
c a l l s them u l t r a b a r r e l l e d s p a c e s ) . S e e D e f i n i t i o n 8 ,
[ 9 5 ] , p a g e 1 0 . H i s P r o p o s i t i o n 5 shows t h a t a n y s p a c e w i t h Banach-Steinhaus
p r o p e r t y i s u l t r a b a r r e l l e d i n t h e s e n s e of
R o b e r t s o n [ 7 8 ] . Hence, i t i s n a t u r a l t o a s k w h e t h e r
the W.
t h e Banach-
S t e i n h a u s p r o p e r t y i m p l i e s b a r r e l l e d n e s s i n t h e s e n s e o f Cefinit i o n 2.35 f o r s p a c e s o v e r v a l u e d d i v i s i o n r i n g s general.
(F,1 .
1)
in
The f o l l o w i n g r e s u l t shows t h a t t h i s i s indeed the case.
THEOREM 2.81:
Let ( E , T ) be a
diwihion hing (F,
I I).
TVS
owch a n o n - t h i w i a & ? y v a l u e d T h e BalLuwing ahe e y u i u a L e n t :
(a)
(E,T) i d bahkelled;
(b)
e a c h p o i n t w i n e bounded n e t H 0 6 c o n t i n u v u d Rincah mappingb 06 ( E , T ) i n t o a T V S ( G , v ) i i 6 equhconRinuoUn.
64
PROLLA
By Theorem 2 . 5 8 , (a) i m p l i e s ( b ) . C o n v e r s e l y , l e t ( E , r ) b e a T V S s a t i s f y i n g c o n d i t i o n (b) above. W e c l a i m t h a t (E,-r) PROOF:
i s b a r r e l l e d . The p r o o f of t h i s c l a i m i s due to Waelbroeck [ 9 5 ] . Let V = ( Vn ) be a -r-closed s t r i n g i n ( E , - r ) . The i d e a o f t h e p r o o f i s t o c o n s t r u c t a T V S (G,w) (W,)
w i t h some fundamental sequence
of v-neighborhoods of t h e o r i g i n , a n d a p o i n t w i s e
family
H of c o n t i n u o u s l i n e a r mappings
:E
+
G
bounded
i s s u c h a way
that
NOW, b y P r o p o s i t i o n 2 . 5 4 ,
c o n t a i n s a -r-neighborhood
Vk
s i n c e , by (b), H i s e q u i c o n t i n u o u s . Thus V
of
0
is a -r-topological
string. Choose and f i x m > n
there exists
fundamental system
ho E F*
I ho I
with
such t h a t
< 1
.
Vm C XoVn
s u c h t h a t g i v e n Vn Choose a n d
B of b a l a n c e d -r-neighborhoods of
a
fix
0 in
E.
R , c h o o s e a T - t o p o l o g i c a l s t r i n g u = (Un) such C IOUn. L e t I b e t h e s e t of a l l such t h a t U1 = U and 'n+l s t r i n g s . The s p a c e G i s t h e a l g e b r a i c d i r e c t sum of t h e f a m i l y { E i ; i E I ) where, f o r e a c h i E I , E . = E . For each k E IN, For each
U E
l e t us define
1
x = ( x1. 11. E I i n by d e f i n i t i o n i f , and o n l y i f Wk C G .
For
xi
G we say t h a t x
E
Wk
i Uk + Vk
E
if
.
i i = (UnInEm
CLAIM:
W
(a) PROOF: L e t if
(Wk) i n a n t t r i n g i n
=
Each
x
wk
E Wk
x = (xi)i t I
.
in and Now
G.
baLanccd and a b d o h b i n g ;
Ih(
Axi
5 1 b e g i v e n . Then 6
Uk
+ Vk
I
if
Xx = ( A x i l i E I
i = (Un)
,
because
65
TOPOLOGICAL VECTOR SPACES
are b a l a n c e d , and
both
Uk
and
that
Wk
is balanced.
L e t now
Vk
x = (xi)i EI
i s absorbing, {xi
1
, ... , x . 1
'k+l Let
PROOF:
i
x
x
Let
x = y
+
z
Uk+l
+
Vk+l
y and
with
z in
i = (U 1 .
if
Wk+l
Hence
n
there exists
Vn
if
E Wk
for
m
.
Then
yi
and
xi
E Uk
+ vk
and
?
for all
x. =O
{ i l ,..., i s } C I .
f o r a f i n i t e set
"k
k E IN.
6otr cc1P
b e g i v e n . Then
E Wm
X -1x
for
implies
+ vk ,
c U;
E Vk
0
=
i
1x1 1. 6
j T h i s shows t h a t
wk ,
c
.
XoVn
x
i s absorbing.
Wk
W e know t h a t g i v e n
PROOF:
vm c
< m.
X -1xi
.
Wk
E
i < j
'k+l
+
belong t o
so
Hence
and so
6
I t follows
c I. Since
{il,...,im1 such t h a t
6 > 0
.
AVk
for a l l
1x1 5
all
C
m
i. = (Ui),
I
there exists
.
Vk
G be g i v e n . W e h av e
i E I, except f o r a f i n i t e set
all
z
in
+
xi E Uk
For e a c h
such
n
i E I,
15 j
2s
that except
w e have
i
xi, E Umj 3 if
=
i i
i
j url+l
= ( C
i.
~~. 1
~
~
i. AoUn3.
we can w r i t e
N ) O W~
Hence
wm
x.
m > n E
This ends t h e proof t h a t Consider i n string
the
G
"m m
implies
i. XoUn3
2
+ XVn a n d 3 and ( c ) is t r u e w i t h
1.
C how,
+
W = (WkIk
T VS t o p o l o g y
v
~
n
+
1, so
Aolx E Wn
X = .A
.
Then
.
is a s t r i n g i n
g e n e r a t e d by t h e
G.
single
W.
For each
i
E
I, let
n1 . :E
+
G
be t h e c a n o n i c a l e m b e d d i n g
66
PROLLA
Notice t h a t f o r e a c h
E . +. G. 1
u; +
Vk
i i = (U )
n n E W ' Since
if that
E d:(E;G).
IT.
1
k
E
c
?Ti-1 (Wk)
i s a -r-neighborhood of
Uk
The f a m i l y
bounded, b e c a u s e e a c h
IN,
H =
-
IT.i I '
0,
w e see
is pointwise
t I}
i s a b s o r b i n g . I n d e e d , g i v e n x E E there
Vk
e x i s t s some 6 > 0 ( d e p e n d i n g on x ) s u c h t h a t I h / > 6 implies x t AV C o n s i d e r now t h e s e t B = { n i ( x ) : i E I } . L e t b B. k '
Then
b
( x ) f o r some
= TT.
i
0
0
if
j
if
io , i f
#
io = ( U n ) n
for each
~~. Thus
C
E G
Hence
XWk
.
bi
c
x t v = Vk
Ut
t AVk C
w + vk ,
.
ri(x)
Thus
because
(X
+
X(Uk
+ Vk),
I t remains to prove t h a t
be such t h a t
W. Then
C
and b . = 0 , 3
x
=
0
7ii(x)
W be a symmetric 7-neighborhood
i E I such t h a t
+ vk
x
B
bi 0
b = (bi)iEI.
k E IN. L e t
i E I. L e t
c u;
E I , and s o
Wk
E
of
E.
x E Uki
a n d so
W) n Vk # @,
Wk , f o r a l l
E
0 in
This
Choose
shows
+ VkC that
i s -r-closed.
Vk
T h i s e n d s t h e p r o o f of Theorem 2 . 8 1 . A similar n o t i o n t o t h a t of a b o r n o l o g i c a l space
(E,r)
over
a t o p o l o g i c a l f i e l d ( F , T ~ w) a s c o n s i d e r e d i n N a c h b i n [ 6 2 ] , 5 8 . To e x p l a i n h i s d e f i n i t i o n w e h a v e t o e x t e n d t h e d e f i n i t i o n o f a bounded s e t g i v e n i n D e f i n i t i o n 2 . 1 6 , f o r s u b s e t s of a T VS ( E , T ) o v e r a v a l u e d d i v i s i o n r i n g ( F , 1 1 ) t o t h e c a s e o f a to-
-
p o l o g i c a l f i e l d ( F , . r F ) , o r more g e n e r a l l y , a t o p o l o g i c a l d i v i s i o n r i n g (F,-rF). DEFINITION 2 . 8 2 :
L e t ( E , T ) b e a T V S o v e r a t o p o l o g i c a l division
ring ( F , T ~ )A . subset -r-neighborhood V
of
0 in
W of
B
C
0 in
F such t h a t
E E,
i s s a i d t o b e b0unde.d i f , g i v e n a
there exists
VB C W.
a
T
F
-neighborhood
Any f i n i t e s e t i s bounded a n d a s u b s e t of a b o u n d e d s e t
is
67
TOPOLOGICAL VECTOR SPACES
bounded. When t h e t o p o l o g y
a subset
is not the discrete
T~
topology,
then
i s b o u n d e d i n (E,T) i f , a n d o n l y i f , f o r e v e r y
B C E
X
of
0
in
When t h e t o p o l o g y
T
i s t h e metric t o p o l o g y d e f i n e d by
?-neighborhood
W
there exists
E,
E F*
such t h a t
XB C W .
A
absolute value coincide.
1x1
+
F
on
then Definition 2.16
F,
Indeed, suppose
B
W a 7-neighborhood
i s bounded i n t h e
E
C
sense
of
there i s sorre
Let
6 > 0
such t h a t Ihj > 6 i m p l i e s B C AW. L e t V = { u G F ; l l ~ l < 6 - l ) . E V, and # 0 , t h e n 1l-l -1 I > 6 , a n d s o B C u - ~ W , i . e .
If
Hence VB
t h e r e i s some
2.82,
Choose
Let
6
Hence
= E
A
-1
-1
.
E
Then
E V
T
> 0
Let
By 2 . 1 6 ,
E.
B C E is b o u n d e d
W be a 7-neighborhood
F- n e i g h b o r h o o d
of
V
of
in
0
0 in
E . By
such t h a t
F
such t h a t
6 > 0
and
0
Conversely, suppose t h a t
C W.
i n t h e s e n s e of 2.82. VB C W.
in
2.82
2.16.
pB C W.
of
and
an
X
1x1
and
-1
> 6 in
i.e.
B C W,
F implies
<
E.
B C XW.
P r o p o s i t i o n 2 . 1 7 and 2 . 1 8 r e m a i n t r u e i f
is a
(EJ)
T VS
over a t o p o l o g i c a l d i v i s i o n r i n g ( F , r F ) a n d " b o u n d e d " i s
meant
from a
T VS
i n t h e s e n s e of D e f i n i t i o n 2 . 8 2 . Also, (E, T
~
e v e r y c o n t i n u o u s l i n e a r map
i )n t o a n o t h e r T V S
(GI T
~
, ) over
T :E
G
--t
t h e same t o p o l o g i c a l divi-
s i o n r i n g (F,rF), maps b o u n d e d s e t s i n t o b o u n d e d s e t s .
For a n y T V S t o p o l o g y
T
on a v e c t o r s p a c e
L ( T ) t h e s e t o f a l l bounded s u b s e t s . C L(-rl).
q i e s on
If E,
{-ri ; i E I } and
T
a T V S t o p o l o g y on
L e t now
8
be
If
i s t h e supremum o f E
T~
i s a non-empty {
C
E,
w e d e n o t e by
r 2 , then
L(-r2)C
f a m i l y of T V S toploT ; ~
i E I ) , then
T
is
and
any
family
of
subsets
of
E,
and
let
68
PROLLA
be t h e f a m i l y of a l l T V S t o p o l o g i e s o n E s u c h that
{
T ; ~i E
B
C L ( T ~ ) S . ince
B C T
E
I}
{@,
E l
i s a T V S topology on
i s bounded, t h e f a m i l y
( R ) = sup
and t h e r e f o r e
i s non-empty.
i E I}
be any T V S t o p o l o g y on
I
Conversely, i f
L(T) 3 B,
then
only i f
T
E.
If
PROPOSITION 2.83:
belongs to
T
on
El
y.
d
r
The topoeogy
{ T ;~i t I]
i b .the d i n e b x
L ( T ) = L(.ri).
it contains
Let
L e t now
Hence
T
id,
topologies
T h i s f a m i l y i s non-empty,
to t h e family
p c
On E A U C ~ and onLg
T V S t t a p o e u g y o n E wkich
7 b e t h e supremum of
belongs
T
E.
{
If
{ T ;~i
T ; ~
L(T)
f
7, t h e n
because
i E I). Now
I}
I I C T V L J C S .
C o n v e r s e l y , if
and
T.
p be a n y T V S t o p o l o g y o n
v p
-
b e t h e f a m i l y of a l l T V S
on E such t h a t T.
i t I}
b e a T V S o v a a totapuLogicd d i u i b i o n
1eZ ( E , T )
hac, .the 4amc b o u n d e d b c . t b a6 PROOF: Let
{Ti;
~ ( 8 ) if,
T C
unique T V S t o p o t a g y R h a i , d o & a n y T V S X v p u k o g y 1-1 o n E , L ( T ) C L ( p ) C
T ( B ) , then
T C
L(T) 2 8.
h i n g (F,T~).T h e h e e x i b t b
id, u
Let
C ~ ( 6 ) Hence . t h e following is true:
T
For a n y T V S t o p o l o g y
(2)
T ; ~
I } . By t h e p r e c e d i n g r e m a r k , w e h a v e
{ T ~ ;i E
L e t now
{
such thatany
E
L(T) = L(r) C L(p).
c
L(V),
and
then
then
TOPOLOG I C A L VECTOR SPACES
69
( N a c h b i n [ 6 2 1 , 5 8 ) : A T VS t o p o l o g y -r o n E i s T = T, where T i s t h e f i n e s t T V S topolo-
DEFINITION 2 . 8 4 :
s a i d t o be b t h U M g i f gy o n
E which h a s t h e s a m e bounded s e t s a s
B of s u b s e t s of
For e v e r y f a m i l y
T.
r(B) is
t h e topology
E,
s t r o n g . I n d e e d , by (1) a b o v e ,
And t h e n b y ( 2 ) a b o v e ;
- c ( B ) i s t h e n c a l l e d t h e AtAong t o p o l o g y genQncLted
The t o p o l o g y
by
8.
THEOREM 2.85: L e t ( E , T ) b e u T V S o v e h u L o p a l a g i c a R d i w i n i o n hing (F,-rF). The I;o&&owing ahe e q u i v a l e n t :
(a)
T
(b)
e w ~ h ybounded L i n e a h m a p p i n g de6ine.d an (E,r) i n c o n t i n -
La a A L h U M g
UUUA
*
PROOF: ( a ) X
C
E
that
u
T : (E, T)
+
(E*,r*)
b e t h e T VS t o p o l o g y on
belongs t o -1
X = T
Lopo&ogy;
.
(b): L e t
mapping. L e t
TVS
p
be a bounded
i f , and o n l y i f t h e r e e x i s t s
(Y).
(which w i l l i m p l y t h a t
p C T
such
Y E T*
C l e a r l y , T i s c o n t i n u o u s from ( E , p ) i n t o ( E * , T * ) . that
linear
E d e f i n e d b y saying that
To p r o v e
i s c o n t i n u o u s from ( E , T )
T
i n t o ( E * , T * ) ) i t s u f f i c e s , by P r o p o s i t i o n 2 . 8 3 and ( a ) , t o show that
L(T)
C L(p).
A C E
L e t then
belong t o
( E , T ) i n t o ( E * , T * ), t h e n A C T-I(B)
T-l(B)
E
,
t o prove t h a t
L(p). Let
d e f i n i t i o n of
p,
(E*,T*) such t h a t
L ( T ) . Since
T(A) = B A E L(p)
T i s bounded
belongs t o
,
it suffices
V be a n open neighborhood of
t h e r e e x i s t s an open V = T-l(W).
Since
L(-r*).
Since
t o show
that
0 i n ( E , p ) . By
neighborhood T(A) = B
from
of
0
in
i s -c*-bounded,
70
PROLLA
t h e r e e x i s t s a TF-neighborhood Hence
UTdl(B)
(b)
=$
i
T-l(W)
(b), T
T
= T,
E* = E ,
REMARK:
0 in
T* =
= L(T), T
L (7
i s continuous, i e. and
of
,
F s u c h t h a t UB C W .
t L(u)
Tel(B)
and
= V,
(a): C o n s i d e r
i d e n t i t y map. S i n c e
U
-
T,
,
as c l a i m e d .
and l e t
T
be
the
i s a bounded l i n e a r map. By
since T c
T c T.
1
T i s always t r u e ,
is strong.
T
Theorem 2 . 8 5 a n d i t s p r o o f a r e d u e t o L .
N a c h b i n (see
8 4 ) . A s a consequence w e
Theorem 8 , Nachbin [ 6 2 1 , p g .
have t h e
following COROLLARY 2 . 8 6 :
divi.5iun king
(b)
Let
(F, I
in a
T
- I).
( E , T ) be
a
T V S
o v e h a n o n - , t h i v i a L L y valued
The dollowing axe cquivaj~en,t:
T VS
h.thong
topohgy.
L e t u s g i v e some e x a m p l e s of s t r o n g T VS t o p o l o g i e s . W e f i r s t
r e c a l l t h a t a T VS ( E , T ) i s c a l l e d locaLLy bounded i f every p o i n t h a s a bounded n e i g h b o r h o o d , o r e q u i v a l e n t l y , i f t h e o r i g i n
has
a bounded n e i g h b o r h o o d .
Id T~ i n n o t t h e d i n c h e t e t v y o e v g y afid ( E , T ) i h a R o c a e L y bounded T V S v v e k ( F , r F ) , t h e n T in a n t h u n g T V S
EXAMPLE 2 . 8 7 :
top0 eogy. PROOF: L e t W
of
T : (E,T)
+
(E*,T*)
b e a T*-neighborhood 0 i n (El
such t h a t
T)
.
Then
AT(V) C W
p o l o g y ) . Now
T(AV)
of
0.
b e a bounded l i n e a r mapping. Let V b e a bounded n e i g h b o r h o o d
Let
T ( V ) i s T*-bounded
toF is not the discrete AV i s a T-neighborhood o f 0 i n E .
(recall that C W,
and
a n d t h e r e e x i s t s h, E F *
T
(Nachbin [ 6 2 ] , p . 85 - 8 6 ) : L e R ( F , T ~ )b e a topeL o g i c a L d i v i b i o n k i n g d u c h t h a t -rF i n n o t $he d i h c k e t e t o p o e o g y and -rF has a c o u n t a b l e BundamentaL n y n t e m v d neighboahaodn at ,the o a i g i n . L e t ( E , T ) b e a T V S O U e h ( F , r F ) . 16 T h a a coun,tizbLe
EXAMPLE 2 . 8 8 :
71
TOPOLOGICAL VECTOR SPACES
at the Ohigin, then
T
i n a
r e s p e c t i v e l y {Un; n = 1 , 2 , 3 ,
...1,
d u n c i a m e n t a l ? n y n t e t n ad n e i g h b o h h o a d n
LopoLogy.
T VS
ntriong
PROOF: L e t
.. I ,
{Vn;n = 1 , 2 , 3 , .
b e a f u n d a m e n t a l s y s t e m o f n e i g h b o r h o o d s a t t h e o r i g i n of ( E J ) , respectively ‘n
’ ‘n+l
that
( F , - r F ) . W i t h o u t loss o f
for a l l
n.
.
L(p) = L (T)
E.
y
b e a n y T VS t o p o l o g y
W e claim t h a t
a s t r o n g topology. in
Let
Let
W
u
n’
v
C
T,
on
assume such
E
w h i c h shows t h a t
T
is
b e any u - n e i g h b o r h o o d of t h e o r i g i n
Suppose t h a t f o r e v e r y
there exists a pair
g e n e r a l i t y w e may
n, with
n’
UnVn
p
i.e.,
W,
un t Un
and
f o r every v
.
E Vn
n
n,
, such
u v 9 W . The s e t A = {v ; n = 1 , 2 , . . I i s -r-bounded. n n n I n d e e d , l e t V b e any ? - n e i g h b o r h o o d of t h e o r i g i n i n E . By con-
that
t i n u i t y a t t h e o r i g i n of t h e mapping ( A , x ) neighborhoods IVlIV2l”’
and
UN
‘VN-l
A
such t h a t
Ax
there
Uvn C V ,
i s -r-bounded,
we can f i n d i n t e g e r
for a l l
1< n 5 N
as c l a i m e d . Hence
k such t h a t
A
UkA C W .
exist
The f i n i t e s e t
UNVN C V.
i s bounded and a n e i g h b o r h o o d
}
be found such t h a t
and so
VN
+
U
of i n
- I.
is
F can
Then
y-bounded,
and
uk v k E W ’
Therefore
which c o n t r a d i c t s t h e a s s u m p t i o n t h a t u v 9 W , f o r a l l n . Hence n n UmVm C W f o r some m . S i n c e T is n o t t h e d i s c r e t e topology, there is
X # 0
a T-neighborhood
with
X E Urn
.
F
Hence
XV
m
C
W
and t h e n
o f t h e o r i g i n . T h i s shows t h a t
C
W
is
T.
NOTES AND REMARKS Most o f t h e c o n t e n t s o f C h a p t e r 2 up t o 2 . 2 8 c a n i n many books o n T V S ; see, f o r e x a m p l e , C h a p t e r I
[111 , o r s e c t i o n s 1 t o 6 o f C h a p t e r I o f S c h a e f e r
of
be
found
Bourbaki
[81]
P r o p o s i t i o n 2.28 r a i s e s t h e f o l l o w i n g q u e s t i o n : w h a t i s t h e w i d e s t c l a s s o f Hausdorff t o p o l o g i c a l d i v i s i o n r i n g s t h a t
can
b e u s e d a s s c a l a r s f o r T VS p r e s e r v i n g t h e p r o p e r t y t h a t l i n e a r f u n c t i o n a l s a r e c o n t i n u o u s i f , and o n l y i f , t h e i r
kernels
are
72
PROLLA
c l o s e d . T h i s w a s a n s w e r e d by N a c h b i n [611. DEFINITION 2.89:
L e t ( F , T ~ b) e a H a u s d o r f f t o p o l o g i c a l d i v i s i o n
r i n g . A Hausdorff topology
w i t h henpeck
06
on
T*
F is s a i d t o be
A Hausdorff t o p o l o g i c a l d i v i s i o n r i n g
d-tXiciiey minimal i f
admiabibLe
( F , T * ) i s a Hausdorff T V S o v e r ( F , T ~ ) .
if
T~
T~
(F,-rF) i s s a i d t o
i s t h e o n l y Hausdorff topology
which i s a d m i s s i b l e w i t h r e s p e c t t o
T
be
on
F
F'
From Theorem 2 o f N a c h b i n [ 6 1 ] and K a p l a n k y ' s c h a r a c t e r i z a -
i t f o l l o w s t h a t any n o n - t r i v i a l l y
t i o n of v a l u e d d i v i s i o n r i n g s
v a l u e d d i v i s i o n r i n g (F,I * I ) i s s t r i c t l y m i n i m a l .
(Nachbin [ 6 1 1 ) : L t . t ( F , T ~ )be. a Haudolrzjd i x p o l o g i c d d i v i n i o n k i n g , T h e dollowing a x t e q u i v a l e n t :
THEOREM 2 . 9 0 :
(a)
(F,-rF)
i n 4 Z h i c R L y mi.nimul;
PROOF: S u p p o s e t h a t ( F , . r F ) i s s t r i c t l y m i n i m a l , and l e t b e a H a u s d o r f f T VS o v e r ( F , T f
functional. If
~ ) .L
is continuous,
i s c l o s e d i n ( F , T ~ ) .I n d e e d , f-l(O)
then
= El
f
a e E
t i n u o u s . Suppose
given a r b i t r a r i l y , l e t saying t h a t
Y
C
belongs t o
T*
to
T
such t h a t
T VS over
f
i s onto
a
linear
{O]
in
(E,T).
# 0.
F t h e topology T*
F,
A
If
f(x)
€
A.
=
conF
defined
T*
is
Hence
i f , and o n l y i f ,
by
f - 1 (Y)
i t i s easy t o see t h a t Y
C
F
E
belonging
f ( X ) = Y . We claim t h a t ( F , T * ) i s a
Hausdorff
( F , . r F ) , i . e . T*
(Alp)
f (a)
= Af(a)-la. Then
i f , and o n l y i f , t h e r e e x i s t s
(see D e f i n i t i o n 2 . 8 2 ) (i)
be
f-l(O) i s closed
i s such t h a t
x
belongs t o
F
T, Since
F
i s i d e n t i c a l l y z e r o and o b v i o u s l y
f ( E ) = F. L e t u s c o n s i d e r o v e r
belongs t o
+
i s c l o s e d because
( F , - r F ) i s H a u s d o r f f by h y p o t h e s i s .
C o n v e r s e l y , assume t h a t t h e k e r n e l If
f : E
et
f-'(O)
(E,T)
+
h
+
.
X
C
i s admissible with r e s p e c t
i n continuoun,
to
TF
73
TOPOLOGICAL VECTOR SPACES
PROOF: L e t ( a , a ) E F x F , and l e t a
+
f3 i n
( F , T * ) . Since
such t h a t f ( a ) =
c1
f
there e x i s t (a,b)
F,
f ( b ) = 8. Now
and
i t follows t h a t
W E T*,
is onto
W be a n open neighborhood of
a
c a n f i n d T-open n e i g h b o r h o o d s A o f a and such t h a t
A
+
B C f-l(W). L e t
(resp. V) i s a
u+vcw.
open
B of b , r e s p e c t i v e l y ,
{O)
Since
the
division
i s r*-closed.
*
(a)
(b)
F.
E
f ( E \ N) = F \ (01,
Now
*
ring
is strictly
(F,-rF)
f
i s c o n t i n u o u s from ( E , T )
minimal,
onto
(E,r)
(F,rF).
( a ) : Suppose ( F , T ~ i) s n o t s t r i c t l y m i n i m a l . L e t
(X,u)
( F , - r F ) . Hence
(F,T*) i n t o ( F , T * ) . P u t t i n g
(F,.rF) x
i d e n t i t y mapping that is
T*
C
rF
.
A
-+
+
Xu
k e r n e l , namely
Hausfrom
see t h a t
we
( 0 3 , because
d e f i n e d by
the
X i s c o n t i n u o u s from ( F , T ~ )i n t o ( F , T * ) ; -r* # T~ , by h y p o t h e s i s , t h e n A X
Since
-+
T*
i s Hausdorff.
a closed
Hence
f a l s e f o r t h e H a u s d o r f f T V S ( F , - r * ) and t h e l i n e a r F
T*
i s continuous
= 1,
i s n o t c o n t i n u o u s from ( F , r * ) i n t o ( F , r F ) , b u t i t h a s
REMARK:
and
(b).
dorff TVS over
+
N =
i s T-open, and
N
be a n o t h e r Hausdorff topology on F s u c h t h a t ( F , T * ) i s a
f : F
that
Since
Now f i s c l e a r l y c o n t i n u o u s as a mapping f r o m .
~
o n t o ( F , T * ) , and s o Hence
6 ) such
U
i s -r*-closed.
{O}
T* = T
i s -r*-open i n
f(E\ N )
V = f ( B ) . Then
a (resp.
f - l ( O ) i f c l o s e d i n ( E , T ) , i t s complement
therefore
so
we
E. Therefore
(i).
PROOF: I f s u f f i c e s t o show t h a t =
x E
b E f - l ( W ) , and since
U = f ( A ) and
n e i g h b o r h o o d of
PROOF: S i m i l a r t o t h a t of
+
a s -r-open i n
f-'(W)
E E
(b)
is
functional
f f X ) = A.
If f o l l o w s f r o m t h e c h a r a c t e r i z a t i o n o f t h e t o p o l o g y
T*
74
PROLLA
g i v e n i n t h e proof of
(a) * ( b ) above, t h a t
only continuous b u t a l s o open, i . e . f (A) A
C
E
belonging t o
T.
f :E
+
is
F
for
all
Hence t h e f o l l o w i n g r e s u l t i s t r u e
(see
Nachbin [ 6 2 1 ,
p. 7 7 ) :
THEOREM 2 . 9 1 :
L e t ( F , T ~ )b e a b t h i c t e y
i s TF-open
not.
minOrid
diuh-
Haundohad
k i n g , and L e i ( E , T ) be a ffaubdohdd T V S v u e h ( F , T ~ ) T. h e n e v e k y n v n - z e k o c o n i i n u v u h L i n e a h duncZionaL f : E + F i b an vpen mapping dkom E o n t o F. biVn
L e t u s now see t h a t
for a strictly
minimal Hausdorff
p o l o g i c a l f i e l d , a " c l o s e d graph theorem" c a n
be
proved
tofor
linear functionals. ( P r o p o s i t i o n 1 9 , N a c h b i n 1 6 2 1 , p . 7 8 ) : LeL (F,TF)
THEOREM 2.92:
divinion king, LcZ ( E , T ) be a Haubdvtr6d T V S v u e k ( F , T ~ )and L e i f : E F b e a LLneatL Aunct i v n a L . T h e dullowing a t e e q u i v a l e n t : b e a b t k i c t l g minima[ Haubdvtdd
-+
(a
f
i b cvntinuvub;
(b
f
hub cLobed g h a p h .
PROOF : S i n c e ( F , T ~ i) s H a u s d o r f f , Conversely, suppose t h a t
(a)
f :E
w i t h c l o s e d graph. L e t us d e f i n e
+
F
E
vector space
E x F, and t h e k e r n e l o f
C o n s i d e r i n g on
E x F
is a +
linear functional by
F
g is a l i n e a r functional onthe g i s t h e graph
t h e product topology, E x F
Hausdorff T V S over (F,-rF). Since
(b) is clear.
g :E x F
for a l l (x,x)
E x F. Clearly,
*
f ( x ) = g(x,O), for a l l
By Theorem 2 . 9 0 ,
x
E
g
of
becomes
f. a
i s continuous. f i s con-
E , o n e sees t h a t
tinuous. S t r i c t l y minimal t o p o l o g i c a l d i v i s i o n r i n g s
are
important
f o r a f u r t h e r r e a s o n , namely t h a t e v e r y f i n i t e - d i m e n s i o n a l s u b -
space of a t o p o l o g i c a l v e c t o r space
over
a
strictly
minimal
75
T O P O L O G I C A L VECTOR SPACES
c o m p l e t e t o p o l o g i c a l d i v i s i o n r i n g i s c l o s e d . T h i s f o l l o w s from Theorem 7 o f Nachbin [61]: THEOREM 2.93:
a Haun-
Nachbin 1 6 1 1 ) : L e t ( F , T ~ b )e
(Theorem 7,
dohdd t o p o C v g i c a L d i v i o i o n k i n g . T h e d o L C o w i n g a h e e q u i u a l e n z : (a)
( F , r F ) i b n t h i c t l y minimae and CompLete.
(b)
Eve4y ~ i n i t e - d i m e n n i o n a B u e c t o h Apace E o u e h F has onCy o n e t o p o l o g y T nuch -that ( E J ) i n a Haundohdd T V S vwek ( F , T ~ ) .
L e t ( F , T ~ b) e a n t t r i c t l ? y m i n i m a L compCete R o p o BogicaC d i v i n i o n h i n g , and L e t ( E , T ) b e a Haubdohda T V S o w c ~ h ( F , - r F ) . T h e n ewehy dinite-dimenniunaC nubnpace 0 6 E i n c C o n e d . COROLLARY 2 . 9 4 :
Since every b a r r e l l e d T V S is quasi-barrelled, t o ask i f t h e c l a s s of b a r r e l l e d spaces i n t h e c l a s s of q u a s i - b a r r e l l e d a d a p t e d from T u r p i n
is
i t i s natural
properly contained
s p a c e s . The f o l l o w i n g
example,
[ 8 9 ] , shows t h a t t h e i n c l u s i o n i s i n d e e d a
proper one. EXAMPLE 2.95:
ring, let
E
Let
(F,
1 1)
vector. L e t
f i n e s t T V S t o p o l o g y on
PROOF: T.
Hence
(E,T)
T~ T
a n o n - t r i v i a l l y valued
division
be t h e s p a c e o f a l l f i n i t e s e q u e n c e s and l e t
be t h e nth-unit
CLAIM I :
be
E f o r which
B
Denote by
T
en the
i s bounded.
i n bofinoBogicaB, h e n c e quani-6ahheLBed.
is stronger than B = T
CLAIM 11: ( E , T )
B = { e n ; n E IN 1 .
.
T
a n d h a s t h e s a m e bounded setsas
in n o t b a 4 h e L l e d .
PROOF: D e f i n e c o n t i n u o u s l i n e a r f u n c t i o n a l s
k 9k(x) =
x
j =1
XjCj
pk : E
+
F
by
76
PROLLA
if
x
=
(6.) 1
where {X.) i s a s e q u e n c e i n
E E;
F with
IX.1
a.
+
3 c o n v e r g e s p o i n t w i s e t o a l i n e a r f u n c t i o n a l P. 7
The s e q u e n c e {P 1 k I f ( E , T ) w e r e b a r r e l l e d , by Theorem 2.59 cp would b e c o n t i n u o u s , and so
v(B) =
would b e bounded, a c o n t r a d i c t i o n .
{A.} 3
I n Definition 2.29, as n
+
a;
( c ) f o l l o w s from (a) and (b),when ln-ll + O
i n p a r t i c u l a r when ( F ,
1. 1 )
i s IR o r C w i t h t h e i r usual
a b s o l u t e v a l u e s . I n d e e d , g i v e n any ho E F*, choose k E IN so that Ik-1 I 5 lXol , a n d t h e n , g i v e n Un , c h o o s e by ( b ) m > n is so b i g t h a t t h e k-fold sum Um + + Um C Un . S i n c e m' -1 -1 -1 b a l a n c e d by ( a ) , x E Urn i m p l i e s k .A x E Urn, and so .A x E UnI
...
which p r o v e s ( c ) The n o t i o n
.
of s t r i n g appeared i n Iyahen
n i t i o n of oupkaDa44eLn and u L t k a b a k k e L ~ (1341 I y a h e n remarks ( [ 3 4 ] , p. 293) t h a t J . W.
[341 i n h i s d e f i -
,
Definition 3.1).
Baker a l s o
considered
t h e n o t i o n of a n u l t r a b a r r e l . According t o 1341
IR or
T V S (E,T) over
sequence ( U such t h a t
n
U1
D e f i n i t i o n 3.1,
exists
a E
t U1
C
B
C is a nup4aba44et i f t h e r e and
'n+l
+
'n+l
for a l l
n '
n
i s c a l l e d a dedining Aequence f o r
ultrabarrel). If
IN.
E
B is closed, it is c a l l e d an uLttaba4ket.
i s bornivorous, B
The
I f each
B.
Un
i s c a l l e d a bornivorous s u p r a b a r r e l
(resp.
B i s a b a l a n c e d convex a n d a b s o r b i n g
subset,
B is a suprabarrel with
then
a balanced s u b s e t B of a
o f b a l a n c e d and a b s o r b i n g non-empty s u b s e t s o f
I f i n addition sequence (Un)
,
2
-n
B = Un
,
n
E
IN,
as a defining
sequence. The n o t i o n o f a non-convex b a r r e l l e d T V S , when C , i s due t o W .
Robertson ( [ 7 8 ], p. 2 4 9 )
, who
F i s IR o r
c a l l e d them UU'UXo v e r IR
6akheLLed. They w e r e d e f i n e d a s f o l l o w s : a T V S ( E , q )
or E,
a:
i s c a l l e d uLtkabatkelLed i f t h e only T V S topologies
i n which t h e r e i s a b a s e o f q - c l o s e d n e i g h b o r h o o d s
o r i g i n , a r e those coarser than
or
q.
P r o p o s i t i o n 2.38,
on
of
the
w i t h F =IR
C , i s due t o I y a h e n ( [ 3 4 1 , Theorem 3 . 1 ) who showed
T V S i s u l t r a b a r r e l l e d i f and o n l y i f e v e r y u l t r a b a r r e l
that is
a a
neighborhood o f t h e o r i g i n .
or
Theorem 2.59 ( B a n a c h - S t e i n h a u s Theorem) , i n t h e case F i s due t o W. R o b e r t s o n ( [ 7 8 ] , Theorem 5 , p . 2 5 0 ) .
C,
=
IR
77
TOPOLOGICAL VECTOR SPACES
Theorem 2 . 4 9 C,
is
( c l o s e d graph Theorem)
When
F = IR o r
,
i n t h e case
Robertson ( [ 7 8 ] , P r o p o s i t i o n
a l s o due t o W. C
Mahowald [ S O ] had
F = IR o r
1 5 , p. 252).
proved t h e f o l l o w i n g
c h a r a c t e r i z a t i o n of b a r r e l l e d l o c a l l y convex spaces: THEOREM 2 . 9 6 :
IR on
C.
L e t ( E , T ) be a R o c a C l y convex Haundoh66 o p a c e o u e h
T h e 6oLLowing a h e e q u i v a l e n t :
(a)
(E,-r)
(b)
d o h e u e h y Banach o p a c e G , t h e 6aLLowing i n Z h u e : a n y L i n e a h m a p p i n g T htrom E i n t o G w h i c h han a c l o n e d g h a p h i n cOM.tiMUOUh.
in b a u e L R e d ;
For a proof see Mahowald [ 5 0 ] , Theorem 2 . 2 ,
p. 1 0 9 .
r e s u l t w a s e x t e n d e d t o non-convex s p a c e s by Iyahen
[34] ,
showed t h a t R o b e r t s o n ' s c l o s e d graph theorem c h a r a c t e r i z e s
This who ul-
t r a b a r r e l l e d spaces: THEOREM 2 . 9 7 :
L e t ( E , T ) be a T V S
UUeh
IR
04
C. T h e ~oUoLLting
atre e q u i u a L e n t : (a) (b)
(E,T) in u l t h a b a h h e L L e d ; h u h euehy c o m p l e t e rneZhic & i n e a h b p a c e G , A h e @?i?vwing t h u e : a n y l i n e a h mapping T 6hom E in-tv G which han a c&oned g h a p h i 0 c o n t i n u o u b .
i d
For a proof see Iyahen [ 3 4 ] , Theorem 3 . 2 , p . 2 9 7 . The n o t i o n of a non-convex
q u a s i - b a r r e l l e d T V S o v e r IR
or
is due t o Iyahen ( [ 3 4 ] , p. 300) , who c a l l e d them quuAi-uUkabahteLLed. Theorem 2 . 6 1 ( c l o s e d graph theorem f o r bounded mapp i n g s ) , i n t h e c a s e F = IR o r C, i s due t o Iyahen ([34] , Theorem 5.1,
p.
301).
I n f a c t , when
F = IR o r
C, more w a s proved. I n d e e d ,
l o c a l l y convex s p a c e s Mahowald ( [ 5 0 ] , Theorem 3 . 1 )
proved
for the
f o l l o w i n g r e l a t i o n between q u a s i - b a r r e l l e d s p a c e s and the c l o s e d graph theorem:
78
PROLLA
THEOREM 2 . 9 8 :
IR
L e t (E,T) be a l o c a l l y c o n v e x Hauddohdd
Apace uvQ.Z
c.. The d o l l a w i n g a t e eyuiwaLenb: quabi-bathellLed;
(a)
(EJ)
(b)
ewetry Banach Apace G , t h e GoLlowing i b t / r u e : a n y bounded l i n e a h mapping 6 t r o m E i n t o G w h i c h hub a c l o n e d gtraph i h c o n t i n u o u d .
ib
60%
For a p r o o f , see Mahowald
[50],
Theorem 3 . 1 , p .
a n a l o g u e f o r non-convex
s p a c e s w a s p r o v e d by I y a h e n :
Let ate. e q u i v a l e n t :
be a T V S
THEOREM 2 . 9 9 :
(E,r)
owP.h
IR
04
109.
Its
C . The d a U o w i n g
(a)
(E,T)
(b)
dux evetry c o m p l e t e m e t h i c l i n e a h Apace G , .the ZjukYowLng i n bhue: a n y bounded l i n e a h mapping d h a m E i n t o G
i b yuabi-u~.thabatrheL&ed;
u h i c h hah a c l o s e d ghaph i n c o n t i n u o u d . F o r a p r o o f see I y a h e n
[ 3 4 J , Theorem 5 . 1 ,
p . 301.
The n o t i o n o f a non-convex b o r n o l o q i c a l T V S , when F i s IR o r C, i s d u e t o I y a h e n ( [ 3 4 ] , D e f i n i t i o n 4 . 1 , p . 298) who called them uL-thabu/rnaLug.icaL. They were d e f i n e d a s f o l l o w s : a T V S ( E , n ) o v e r IR or C is c a l l e d ulthabohnoLogiCaL i f every bounded l i n e a r map from ( E , T ) i n t o a n y T V S o v e r JR o r C i s cont i n u o u s . Our Theorem 2 . 4 2 shows t h a t D e f i n i t i o n 2 . 4 0 and Iyahen's definition a r e equivalent.
The f o l l o w i n g r e s u l t was p r o v e d
by
I y a h e n , and should b e compared w i t h Theorem 2 . 9 9 . THEOREM 2 . 1 0 0 :
Let
(E,T)
b e a T V S oweh IR o h C..
The d u l l o d n g
atre e q u i v a l e n t : (a)
(EJ)
(b)
6 0 4 eve4.y cornpeebe m e t h i c l i n e a h cspace G , t h e do&LoWLng i s t t u e : a n y bounded L i n e a t mapping ( R a m E i n t o G i d cofitinuoun.
i A
ulLttabotrnuLogical;