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Chapter Five Local Completeness
151
CHAPTER FIVE LOCAL COMPLETENESS
5 . 1 D e f i n i t i o n s and c h a r a c t e r i z a t i o n s . D e f i n i t i o n 5.1.1:
L e t E be a space. A sequence (x(n):n=1,2,..)
in E is
s a i d t o be l o c a l l y convergent o r Mackey convergent t o an element x o f E if t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o x i n
b. A
se-
quence i s l o c a l l y n u l l i f i t i s l o c a l l y convergent t o t h e o r i g i n . A sequence i s c a l l e d l o c a l l y Cauchy o r Mackey Cauchy i f i t i s a Cauchy sequence i n
5
f o r a c e r t a i n d i s c B i n E. Lemm 5.1.2:
L e t E be a n e t r i z a b l e space and (x(n):n=1,2,..)
a n u l l se-
quence i n E, t h e n t h e r e i s an i n c r e a s i n g unbounded sequence o f p o s i t i v e r e s l numbers (a(n):n=1,2,..)
such t h a t t h e sequence (a(n)x(n):n=l,Z,..)
con-
verges t o t h e o r i g i n . P r o o f : L e t (Uk:k=1,2,..)
be a d e c r e a s i n g b a s i s o f a b s o l u t e l y convex
0-nghbs i n E. Since (x(n):n=1,2,..)
converges t o t h e o r i g i n , we can f i n d
an i n c r e a s i n g sequence (n(k):k=1,2,. . ) o f p o s i t i v e i n t e g e r s such t h a t 1 x ( n ) € k- Uk, n 2 n ( k ) , k=1,2, ... We s e t a ( n ) : = l i f l L n < n ( l ) and a(n):=k i f
...
n < n ( k + l ) , k=1,2, Then a ( n ) x ( n ) e Uk f o r e v e r y n’n(k), k=1,2, n(k) Thus t h e sequence (a(n)x(n):n=1,2,..) converges t o t h e o r i g i n i n E.
...
//
P r o p o s i t i o n 5.1.3:
(i) A sequence (x(n):n=1,2,..)
converges t o x i f and o n l y i f (x(n)-x:n=l,Z,..) ( i f ) A sequence (x(n):n=1,2,..)
i n a space E l o c a l l y
i s locally null.
i n a space E i s l o c a l l y n u l l i f and o n l y if
t h e r e i s an i n c r e a s i n g unbounded sequence ( a ( n):n=1,2,. numbers such t h a t (a(n)x(n):n=1,2,..)
.) o f p o s i t i v e r e a l
converges t o t h e o r i g i n i n E.
P r o o f : (i) i s t r i v i a l . To prove ( i i ) , i f (a(n):n=1,2,..)
i s an i n c r e a s -
i n g unbounded sequence o f p o s i t i v e r e a l numbers such t h a t (a( n ) x ( n ) :n=I,. . ) converges t o t h e o r i g i n , t h e s e t B : = ZEx(a(n)x(n):n=1,2,..)
i s a closed
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152
d i s c i n E such t h a t the sequence ( x ( n ) : n = l , Z , . . ) converges t o t h e o r i g i n in Conversely, i f ( x ( n ) : n = 1 , 2 , . . ) i s l o c a l l y n u l l , t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o the o r i g i n in EB. Applying 5.1.2 t h e r e i s an incresing unbounded sequence of p o s i t i v e real numbers ( a ( n ) : n = 1 , 2 , . . ) such t h a t ( a ( n ) x ( n ) : n = l , Z , . . ) converges t o t h e o r i q i n i n and
k.
hence i n E .
/I
Proposition 5.1.4:A sequence in a metrizable space i s convergent i f and only i f i t i s l o c a l l y convergent. Proof: I t i s enough t o apply 5.1.2 and 5 . 1 . 3 ( i i ) .
//
Definition 5.1.5: A space is l o c a l l y conplete i f every l o c a l l y Cauchy sequence i s 1ocal l y convergent. Proposition 5.1.6: Let E be a space. The following conditions a r e equiVal e n t : ( i ) E i s l o c a l l y complete. ( i i ) Every closed d i s c in E i s a Banach d i s c . ( i i i ) I f u i s a topology of t h e dual pair(E,E') and B i s a d i s c in E , then every Cauchy sequence i n i s convergent in ( E , u ) . ( i v ) Every bounded subset of E i s included i n a Banach d i s c . Proof: ( i ) - + ( i i ) . Let B be a closed d i s c in E and ( x ( n ) : n = 1 , 2 , . . ) a Cauchy sequence i n g. By ( i ) t h e r e i s x in E such t h a t t h e sequence converges l o c a l l y t o x and hence converges i n E . Since ( x ( n ) : n = 1 , 2 , . . ) is bounded in E B , t h e r e i s a > O such t h a t x ( n ) E a B , n = 1 , 2 , ... Therefore xEaB, B being closed in E . Moreover f o r every b>O t h e r e i s a p o s i t i v e i n t e g e r p such t h a t x ( n ) - x ( m ) ~ b Bf o r every n , m 2 p. L e t t i n g m t o i n f i n i t y we have
5
that x(n)-xEbB i f n
p . T h u s t h e sequence converges t o x i n
%
and EB i s
complete. ( i i ) - - b ( i i i ) . Let u be a topology of the dual p a i r ( E , E ' ) and A a d i s c in E . Set B f o r the c l o s u r e of A i n E. I f ( x ( n ) : n = l , Z , . . ) i s a Cauchy sequenc e i n E A y then i t i s a l s o a Cauchy sequence i n t h e Banach space E B , hence ( x ( n ) : n = 1 , 2 , . . ) converges i n Since the canonical i n j e c t i o n from EB i n t o ( E , u ) i s continuous, i t follows t h a t t h e sequence converges i n ( E , u ) . ( i i i ) - - b ( i ) . Let ( x ( n ) : n = l , Z , . . ) be a Cauchy sequence i n f o r a certain d i s c B . By ( i i i ) t h e r e i s x i n E such t h a t the sequence converges t o x i n ( E , s ( E , E ' ) ) . On t h e o t h e r hand f o r every b > O t h e r e i s a p o s i t i v e i n t e g e r
k.
5
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p such t h a t i f n , m 2 p, then x(n)-x( m)E tRc bC, where C i s the closure of B i n E . Letting m t o i n f i n i t y we have t h a t x(n)-xebC i f n 5 p , and t h u s the sequence ( x ( n ) : n = 1 , 2 , . . ) converges l o c a l l y t o x. ( i i ) - + ( i v ) and ( i v ) - - b ( i ) a r e t r i v i a l .
//
Corollary 5.1.7:
If a space E is l o c a l l y complete, t h e n i t i s l o c a l l y
complete f o r every topology of the dual p a i r ( E , E ' ) . Corollary 5.1.8: If a space i s sequentially complete, then i t is local l y conplete. Proof: Every closed d i s c i n a sequentially complete space i s a Banach d i s c by 3.2.5.
//
Corollary 5.1.9: A netrizable space i s l o c a l l y complete i f and only i f i t i s complete. Proof: Apply 5.1.8 and 5.1.4.
//
Corollary 5.1.10: Every barrel i n a l o c a l l y complete space i s bornivorous. I n p a r t i c u l a r a l o c a l l y complete space i s barrelled i f and only i f i t i s quasi barrel led. Proof: By 3.2.7 b a r r e l s absorb Banach d i s c s .
//
Theorem 5.1.11: Let ( E , t ) be a space. The following conditions a r e equivalent : ( i j ( E , t ) i s l o c a l l y complete. (ii) The closed absolutely convex hull of every l o c a l l y null sequence i n ( E , t ) i s compact. ( i i i ) The closed absolutely convex hull of a null sequence in ( E , s ( E , E ' ) ) i s compact i n ( E , s ( E , E ' ) ) . ( i v ) The closed absolutely convex hull of a null sequence i n ( E , t ) i s compact. Proof: ( i ) - - - ( i i i ) . Let ( x ( n ) : n = 1 , 2 , . . ) be a null sequence i n E endowed w i t h i t s weak topology. Let B be i t s closed absolutely convex h u l l . Since E i s l o c a l l y conplete, EB i s a Banach space, hence we apply 3.2.12 t o obt a i n t h a t B i s compact i n ( E , s ( E , E ' ) ) . ( i i i ) - + ( i v ) . According t o ( i i i ) the closed absolutely convex hull of a
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154
n u l l sequence i n ( E , t )
i s s(E,E')-compact
and t-precompact,
nence K1§18,4,
( 4 ) i m p l i e s t h a t i t i s a l s o compact i n ( E , t ) . ( i v ) - + ( i i ) i s t r i v i a l s i n c e e v e r y l o c a l l y n u l l sequence i n E i s n u l l i n (E,t). (ii)--(i).
5 . We
a Cauchy sequence i n
L e t B be a d i s c i n E and (x(n):n=l,Z,..)
o f positive ink and we s e t y ( k ) : = 2 ( x ( n k t 1 ) - d n k ) ) .
s e l e c t a s t r i c t l y i n c r e a s i n g sequence (nk:k=1,2,..)
t e g e r s such t h a t x( nktl)-x(nk)E C l e a r l y (y(k):k=1,2,..)
2-'%
converges t o t h e o r i g i n i n
5
c l o s e d a b s o l u t e l y convex h u l l A i s compact i n ( E , t ) . '
k z ( p ) : = 5 2 - y ( k ) , p=1,2,..,
hence, by ( i i ) , i t s Now t h e sequence
i s a Cauchy sequence i n ( E , t )
hence t h e r e i s an element x i n E such t h a t (z(p):p=1,2,..) i n (E,t).
c o n t a i n e d i n A, converqes t o x
) - x ( n ), t h e sequence (x(n):n=l,Z,..) converP+l P Thus, a o p l y i n g 5.1.6 (iii), t h e space ( E , t ) i s
Since z ( p ) = x ( n
ges t o x(nl)+x
i n (E,t).
l o c a l l y complete.
Example 5.1.12:
/I L o c a l l y c o n p l e t e spaces which a r e n o t s e q u e n t i a l l y com-
p l e t e . Every F r e c h e t space endowed w i t h i t s weak t o p o l o g y i s l o c a l l y c o r n p l e t e by 5.1.7. I n p a r t i c u l a r ( c o , s ( c o y l 1) ) i s l o c a l l y complete. I f e ( n ) i t i s easy t o see t h a t denotes t h e n - t h c a n o n i c a l u n i t v e c t o r i n c 10, x ( n ) : = e( 1)+...+ e ( n ) , n=l,Z,..,is a s ( c o y l )-Cauchy sequence i n co which
does n o t converge, hence (co,s(co,l
1
) ) i s n o t s e q u e n t i a l l y complete. More
g e n e r a l l y , if E i s a n o n - r e f l e x i v e F r e c h e t space, whose s t r o n g dual i s separable, t h e n (E,s( E Y E ' ) ) i s l o c a l l y complete b u t n o t s e q u e n t i a l l y c o m l e t e . Indeed, t a k e a p o i n t
ZE
E l ' \ E. There i s a bounded subset B of E such
t h a t z belongs t o t h e c l o s u r e C o f B i n (E",s(E",E')). i s separable, (C,s(E",E'))
n=1,2,..)
i n B c o n v e r g i n g t o z i n (E",s(E",E')).
a Cauchy sequence i n (E,s(E,E')) P r o p o s i t i o n 5.1.13:
t e , t h e n E i s l o c a l l y complete. P r o o f : L e t B be a c l o s e d d i s c i n E . The space space and a c l o s e d hyperplane o f
D e f i n i t i o n 5.1.14:
Thus ( x ( n ) : n = l , 2 , . . . ) i s
which does n o t converge.
L e t F be a hyperplane o f
complete spaces and hence
Since (E',b(E',E))
i s t r e t r i z a b l e , hence t h e r e i s a sequence ( x ( n ) :
5 . Thus
E.
I f F i s l o c a l l y comple-
hnF=
FBnF i s a Banach
EB i s complete as a p r o d u c t of
E i s l o c a l l y conplete.
//
L e t E be a space and A a non-void subset o f E . A
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155
point x i s a local l i m i t point of A i f t h e r e is a sequence i n A l o c a l l y convergent t o x. We say t h a t A i s l o c a l l y closed i f every local limit point of A belongs t o A. A s e t B i s l o c a l l y dense i n A i f every point of A i s a local l i m i t point of B . We assume t h a t t h e void set i s l o c a l l y closed. Exanple 5.1.15: Let ( E i : i E I ) be an i n f i n i t e family of spaces and E i t s product. Let x = ( x ( i ) : i ~I ) be an element of Eo. There i s a sequence J:= ( i ( n ) : n = l , Z , . . ) i n I such t h a t x ( i ) = 0 i f i e I ' J . We s e t x n : = ( x n ( i ) : i E I ) defined by s ( i ) = O i f i c I \ J o r i = i ( m ) , m z n , and x,(i(m)) = x ( i ( m ) ) i f m=l, n . We s h a l l s e e t h a t t h e r e i s an absolutely convex compact subset C of Eo such t h a t x ( n ) converges t o x i n ( E o ) C . We take B i :={O\ i f i c I \ J
...,
and B : = a c x ( ( p x ( i ( p ) ) ) ) , p = 1 , 2 , ... Clearly t h e s e t B : = TI ( B i : i € I ) i s i ( P) a compact subset of Eo. On the o t h e r hand i f C is t h e closed a b s o l u t e l y convex hull of ( n ( x n - x ) : n = 1 , 2 , . . ) , i t i s obvious t h a t CcB and x - x n c ( l / n ) C f o r n = 1 , 2 , . . , from where i t follows t h a t C i s a compact absolutely convex subset of Eo and t h a t xn converges t o x i n ( E o I C . i n p a r t i c u l a r we obtain the following Proposition 5.1.16:
If ( E i : i €
I ) i s a non-void family of spaces and E
i s i t s product, then e ( E i : i e I ) is l o c a l l y dense in Eo. Proposition 5.1.17: The i n t e r s e c t i o n of l o c a l l y closed sets i s l o c a l l y closed. Definition 5.1.18: The local closure of a subset A of a space E i s t h e i n t e r s e c t i o n of a l l t h e l o c a l l y closed subsets of E containing A . By 5.1. 17 the local closure of A i s l o c a l l y closed and contains a l l t h e local l i m i t points of A . Definition 5.1.19: A subset A of a space E i s s a i d t o be lo c a ll y comp l e t e i f every local Cauchy sequence i n A converges l o c a l l y t o a point of
A.
Proposition 5.1.20: ( i ) Every l o c a l l y complete subset of a space E i s l o c a l l y closed. ( i i ) Every l o c a l l y closed subset of a l o c a l l y complete space E i s l o c a l l y compl e t e .
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156
I f E i s a space, t h e l o c a l c o m p l e t i o n o f E i s d e f i -
D e f i n i t i o n 5.1.21:
ned as t h e l o c a l c l o s u r e o f E i n i t s completion. I t i s denoted b y ? . 1.20,
? coincides
By 5.
w i t h t h e i n t e r s e c t i o n o f a l l t h e l o c a l l y complete sub-
spaces o f t h e c o m p l e t i o n o f E c o n t a i n i n g E . Observation 5.1.22:
By 5.1.4
e v e r y m t r i z a b l e space i s l o c a l l y dense i n
i t s completion, hence t h e l o c a l Completion o f a m e t r i z a b l e space c o i n c i d e s w i t h i t s completion. Lemma 5.1.23:
L e t E and F be two spaces and f:E-----*F
a continuous li-
near mapping. Then
( i )I f (x(n):n=1,2,..)
converges l o c a l l y t o x, t h e n (f(x(n)):n=1,2,
... )
converges l o c a l l y t o f( x).
(ii)If A i s a l o c a l l y c l o s e d subset of F, t h e n f - l ( A ) i s a l o c a l l y c l o s e d subset o f E. P r o o f : ( i ) i s obvious. To prove ( i i ) , l e t (x(n):n=1,2,..) be a sequence 1 i n f- ( A ) l o c a l l y convergent t o x i n E. By ( i ) t h e sequence ( f ( x ( n ) ) : n = l , Z,..)
i s l o c a l l y convergent t o f ( x ) . Since A i s l o c a l l y c l o s e d , f ( x ) be-
1
l o n g s t o A, t h u s x belongs t o f - ( A )
P r o p o s i t i o n 5.1.24:
-11
L e t f:E------F
be a c o n t i n u o u s l i n e a r mapping. I f
A i s a subset o f E, t h e n t h e image o f t h e l o c a l c l o s u r e o f A by f i s i n cluded i n t h e l o c a l c l o s u r e of f ( A ) . P r o o f : Take any p o i n t x i n t h e l o c a l c l o s u r e o f A and B any l o c a l l y c l o s e d subset o f F c o n t a i n i n g f ( A ) . T h e r e f o r e A i s i n c l u d e d i n t h e l o c a l l y 1 o f E. Then x c f - ( B ) , hence f ( x ) a B and conseauently c l o s e d subset f - ' ( B ) f ( x ) belongs t o t h e l o c a l c l o s u r e o f f ( A ) . / /
P r o p o s i t i o n 5.1.25: mapping f : E - - - - -
L e t E and F be spaces. Given a continuous l i n e a r --&,
F, t h e r e i s a unique c o n t i n u o u s l i n e a r mapping f : E - - - - - F
N
whose r e s t r i c t i o n t o E c o i n c i d e s w i t h f .
-
P r o o f : The uniqueness i s t r i v i a l . On t h e o t h e r hand, g i v e n f t h e r e i s a f i A
continuous l i n e a r e x t e n s i o n t o t h e completions, f : E - - - - + we show t h a t ?(:) f(E)
F. We a r e done i f
i s i n c l u d e d i n ? . To see t h i s , observe t h a t E i s t h e l o -
cal closure o f E i n A
A
?.
By 5.1.24,
;(?)
i s included i n the l o c a l closure o f N
i n F which i s c l e a r l y a subset o f F.
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157
Corollary 5.1.26: Let E be a metrizable space and F a l o c a l l y complete space. Given a continuous l i n e a r mapping f:E-----cF, t h e r e i s a continuous l i n e a r extension ? t o t h e completion of E w i t h values i n F. Proof: Apply 5.1.22 and 5.1.25.
//
Convergence and local convergence coincide i n a metrizable space. Much more can be s a i d . Theorem 5.1.27: Let E be a metrizable space. Then ( i ) ( 1) For every sequence of bounded s e t s (An:n=1,2,. .) there a r e c( n ) > 0, n = 1 , 2 , . . , such t h a t U ( c ( n ) A n : n = 1 , 2 , . . ) i s a bounded subset of E. ( 2 ) For every sequence of bounded s e t s ( A n : n = 1 , 2 , . . ) t h e r e is a closed d i s c A such t h a t each An i s bounded i n EA. ( i i ) For every bounded subset A of E t h e r e i s a closed d i s c 6 such t h a t A i s included i n 6 and t h e topologies induced on A by E and coincide.
5
Proof: Let (U :n=1,2,..) be a decreasing b a s i s of closed absolutely conn vex 0-nghbs i n E . ( i ) ( l ) For every p o s i t i v e i n t e g e r n we determine c ( n ) > O such t h a t c(n)An i s included i n U n . One e a s i l y sees t h a t U(c(n)An:n=1,2,..) i s bounded i n E . ( i ) ( 2 ) Proceeding as i n the former proof i t i s enough t o take a s A t h e c l o sed absolutely convex hull of U ( c ( n ) A n : n = 1 , 2 , . . ) . ( i i ) We can suppose,without loss of g e n e r a l i t y , t h a t A i s absolutely convex. F i r s t we f i n d a closed d i s c 6 i n E containing A such t h a t f o r every a z O t h e r e i s a p o s i t i v e i n t e g e r n w i t h A n U n c a B . Given A t h e r e i s c ( i ) > O such We determine b ( i ) 4 c ( i ) such t h a t t h e seauence that ACc(i)Ui, i=1,2, ( c ( i ) b ( i ) - ' : i = l , Z , . . ) converges t o zero. We set B : = n ( b ( i ) U i : i = l , 2,...). Clearly B i s a closed d i s c i n E containing A. Given a > O there i s a p o s i t i ve i n t e g e r j such t h a t i f i 2 j , t h e n c ( i ) < a b ( i ) , and t h e r e f o r e ACab(i)Ui i f i 2 j . On t h e o t h e r hand n ( a b ( i ) U i : i = l , . . , j - l ) i s a 0-nghb in E , hence
...
t h e r e is a p o s i t i v e i n t e g e r n with Unc ~ ( a b ( i ) U i : i = l Y . . . j - l ) ,from this i t follows t h a t AnUn C n ( a b ( i ) U i : i = 1 , 2 , . . ) = aB. Now t o prove t h a t E and EB induce t h e same topology on A i t i s enough t o show t h a t both induced topol o g i e s have the same b a s i s o f 0-nghbs i n A, by RR,ch6,1,Lemnal. Since t h e topology of EB i s f i n e r than t h e topology of E, t h e conclusion follows from our construction of B
-//
-Corollary
5.1.28:
I f E i s a metrizable space and A i s a preconpact ( r e s p .
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158
compact) subset o f E, t h e n t h e r e i s a c l o s e d d i s c B c o n t a i n i n g A such t h a t
A i s precompact ( r e s p . compact) i n ER. Proof: R e c a l l t h a t n o t o n l y t h e t o p o l o g i e s induced by E and
5
coincide
on A b u t a l s o t h e u n i f o r m i t i e s , by K1,§28.6.(3).//
Theorem 5.1.27
and P r o p o s i t i o n 5.1.2
suggest t h e f o l l o w i n g d e f i n i t i o n s
due t o GROTHENDIECK.
A space E i s s a i d t o s a t i s f y t h e Mackey convergence
D e f i n i t i o n 5.1.29: c o n d i t i o n (M.c.c.)
i f e v e r y n u l l sequence i n
E i s l o c a l l y n u l l . A space E
i s s a i d t o s a t i s f y t h e s t r i c t Mackey c o n d i t i o n ( s . M . c . )
i f f o r e v e r y boun-
ded subset A o f E t h e r e i s a c l o s e d d i s c B such t h a t t h e t o p o l o g i e s induced on A by
E
and
coincide.
Observation 5.1.30: s.M.c.
( i ) By 5.1.27 e v e r y m e t r i z a b l e space s a t i s f i e s t h e
(ii)I f a space s a t i s f i e s t h e s.M.c., ( i ) I f (E,t)
P r o p o s i t i o n 5.1.31: t h e M.c.c.)
t h e n i t s a t i s f i e s t h e M.c.c.
i s a space s a t i s f y i n g t h e s.M.c.(resp.
and F i s a subspace o f E, t h e n ( F , t )
s a t i s f i e s t h e s.M.c.
(resp.
t h e M.c.c.). ( i i ) I f ((En,tn):n=l,2,..)
. ) and
t h e n Ti(( En,tn):n=1,2,.
( r e s p . t h e M.c.c.), t i s f y t h e s.M.c.
i s a sequence o f spaces s c t i s f y i n g t h e s.M.c.
.. )
e(( En,tn):n=1,2,.
sa-
( r e s p . t h e M.c.c.).
P r o o f : (i)l. Suppose
E has t h e M.c.c.
and l e t (x(n):n=1,2,..)
be a n u l l
There i s a c l o s e d d i s c B i n E such t h a t t h e sequence
sequence i n ( F , t ) .
converges t o t h e o r i g i n i n EB, hence C:=Br\F i s a c l o s e d d i s c i n F such converges t o t h e o r i g i n i n Fc. 2 . Now we suppose t h a t
t h a t (x(n):n=1,2,..)
E has t h e s.M.c.
and t a k e a d i s c A i n F. There i s a c l o s e d d i s c B i n E con-
t a i n i n g A such t h a t , f o r e v e r y a,O,
t h e r e i s a 0-nghb V i n E w i t h V n A c a B .
Taking C:=BAF we have t h a t C i s a c l o s e d d i s c i n F c o n t a i n i n g A such t h a t f o r e v e r y a > O t h e r e i s a 0-nghb V ( \ F i n F w i t h V n F A A C a B n F ( i i ) l . We s e t G:= TT ( ( E n , t n ) :n=1,2,. n i c a l p r o j e c t i o n . L e t (x(k):k=1,2,..) (x(k,n):n=1,2,..),
k=1,2
,...
a c l o s e d d i s c B n i n (En,tn) origin i n (E )
. We
. ) and w r i t e p,:G-----E
aC.
m be a n u l l sequence i n G w i t h ~ ( k =)
I f each ( E n , t n )
s a t i s f i e s t h e M.c.c.
such t h a t (x(k,n):n=1,2,..)
s e t B:=TT(nBn:n=1,2,..),
Bn prove t h a t (x(k):k=1,2,..)
=
f o r t h e canothere i s
converges t o t h e
which i s a d i s c i n G. We
converges t o t h e o r i g i n i n GB. Given a > O t h e r e
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159
i s a p o s i t i v e i n t e g e r m with na > 1 i f n , m y and t h e r e i s a p o s i t i v e i n t e g e r s such t h a t x ( k , n ) c aBn i f n = l , . . , m and k 5 s . On the o t h e r hand i f k s and n > m , we have t h a t x ( k , n ) E B n , hence x ( k , n ) E n - l p n ( B ) c p n ( a B ) . Thus i f k s we have t h a t x( k ) C aB. 2. Suppose t h a t each ( E n y t n ) s a t i s f i e s t h e s.M.c. and l e t A be a d i s c i n G. Clearly A c T i ( p n ( A ) : n = 1 , 2 , . . ) . For every p o s i t i v e i n t e g e r n t h e r e i s a closed d i s c B n i n ( E n y t n ) such t h a t p n ( A ) C B n and t h e topologies induced by ( E n , t n ) and t h e normed space generated by B n coincide on p n ( A ) . We s e t B : = V ( n B n : n = 1 , 2 , . . ) , which i s a closed d i s c i n G. Given a > O , t h e r e i s a pos i t i v e i n t e g e r m such t h a t na.1 i f n>m. Now i f n = l , , , , m t h e r e i s a closed absolutely convex 0-nghb Vn i n ( E n y t n ) such t h a t V n ~ p n ( A ) c a B nW . e set V:= 77 ( V n : n = l , . . , m ) x l T ( E n : n = m t l , m + 2 , . . . ) . Clearly V i s a 0-nghb i n G and
VnAcaB. T h u s
G s a t i s f i e s t h e s.M.c.
3. Observe t h a t , according t o ( i ) , ( i i ) 1 and 2 , i f E l ,
...,E P
satisfy the ...,p ) s a t i s -
s.M.c. ( r e s p . t h e M . c . c . ) , then n ( E i : i = l ,... , p ) = Q ( E . : i = l , 1 f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) . 4. I f ( x ( k ) : k = l . Z , . . ) i s a sequence i n @ ( ( E , t n ) : n = 1 , 2 , . . ) converging t o n the o r i g i n , w i t h x ( k ) = ( d k , n ) : n = l , Z , . . ) , k=1,2,.., then there is a p o s i t i v e i n t e g e r m w i t h x(k,n)=O i f n > m and k = 1 , 2 , . . and ( x ( k ) : k = l , Z , . . ) converges to the origin i n €B((Enytn):n=l m). On t h e o t h e r handyif A i s a bounded subset of @ ( ( E n , t n ) : n = l , 2 ,...), t h e n t h e r e a r e a p o s i t i v e i n t e g e r m and bounded subsets A of ( E n , t ) , n = l , . . , m , such t h a t A i s included in
,...,
n
n
@(An:n=1,2,..). Now we can apply 3 t o obtain t h a t i f ( E n y t n ) s a t i s f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) , n = 1 , 2 , . . , then e((E, t ) : n = l , Z , . . ) a l s o s a t i s n n f i e s the s.M.c. ( r e s p . the M . c . c . ) .
//
Example 5.1.32: Uncountable products of spaces s a t i s f y i n g t h e s.M.c. nay f a i l the M.c.c. Let I be t h e s e t of a l l increasing unbounded sequences of p o s i t i v e real numbers. We take E:= T T ( R i : i c I ) , where each Ri i s a copy o f t h e r e a l s . Each i E I i s a sequence ( i ( n ) : n = l , 2 , . . ) i n R . For every p o s i t i v e i n t e g e r n we s e t x ( n ) : = ( x ( n , i ) : i € I ) € E defined by d n , i ) : = i ( n ) - ' , f o r every i c I . Clearly t h e sequence ( x ( n ) : n = l , Z , . . ) converges t o the o r i g i n i n E , b u t i t i s not l o c a l l y convergent, f o r i f i t were t h e r e would be an e l e ment i c 1 such t h a t ( i ( n ) x ( n ) : n = 1 , 2 , . . ) would converge t o the o r i g i n i n E ( 5 . 1 . 2 ) . Then t h e limit of t h e sequence ( i ( n ) x ( n , i ) : n = l , Z , . . ) i n R would be zero, which i s inpossible s i n c e i t converges t o 1.
160
BARRELLED LOCAL L Y CON VEX SPACES
Now we consider the r e l a t i o n of local conpleteness and barrelledness properties. Proposition 5.1.33: Let ( E , t ) be a space such t h a t t = d E . E ' ) . The spac e ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f every null sequence i n E-equicontinuou s . ( E ' ,s( E ' , E ) ) i s Proof: Applying 5.1.11, ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f t h e closed absolutely convex hull of every null sequence i n ( E ' , s ( E ' , E ) ) i s compact, and this i s t r u e i f and only i f every null sequence i n ( E ' , s ( E ' , E ) ) i s an ( E , m ( EYE'))-equicontinuoum.
//
Proposition 5.1.34: If E i s a space whose weak dual is l o c a l l y complete, then every barrel in E i s bornivorous. Proof: Suppose t h a t T i s a barrel in E and B a bounded subset of E not absorbed by T . For every p o s i t i v e i n t e g e r n t h e r e i s dn)eB such t h a t x ( n ) # n 2T. Applying Hahn-Banach's Theorem, we can obtain a sequence ( u ( n ) : n = 1 , 2 , . . ) i n E' such t h a t < x ( n ) , u ( n ) > = n and u ( n ) c ( n T ) : n = 1 , 2 , . . Clearly ( u ( n ) : n = 1 , 2 , . . ) converges t o t h e o r i g i n i n ( E ' , s ( E ' , E ) ) . Since ( E ' , s ( E ' , E ) ) i s l o c a l l y complete, t h e closed a b s o l u t e l y convex hull C of ( u ( n ) : n = 1 , 2 , . . ) i s compact i n ( E ' , s ( E ' , E ) ) , t h e r e f o r e i t s polar V:=C" i n E i s a 0-nghb i n (E,m(E,E')). As B i s a bounded subset of E , t h e r e i s a > O such t h a t B c a V = aC; hence l < x ( n ) , u ( n ) > [ 6 a , n = 1 , 2 , . . , which i s a c o n t r a d i c t i o n .
//
Corollary 5.1.35: A space E i s b a r r e l l e d i f and only if E i s quasibarrel l e d and ( E ' , s ( E' ,E))
i s l o c a l l y complete.
We s h a l l s e e l a t e r ( 8 . 2 ) t h a t t h e local completeness of t h e weak dual i s , in a sense, t h e weakest barrelledness condition.
5.2 S t a b i l i t y of Mackey spaces.
Now we t u r n our a t t e n t i o n t o t h e problem of when a subspace o f a Mackey space i s i t s e l f a Mackey space. More p r e c i s e l y , i f ( E , F ) i s a dual p a i r , t a topology on E compatible w i t h t h e dual p a i r , G a subspace o f E and i f ( E , t ) i s a Mackey space, i . e . , t=m(E,E'), when does ( G , t ) coincide with (G,m(G,(G,t)
')I?
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161
F i r s t observe t h a t , s i n c e t h e p r o p e r t y o f b e i n g a Mackey space is p r e served by separated q u o t i e n t s ( s e e K1,92.2.(
3 ) ) , complemented subspaces of
Mackey spaces a r e Mackey spaces. I n p a r t i c u l a r , c l o s e d f i n i t e - c o d i w n s i o n a l subspaces o f Mackey spaces a r e Mackey spaces. The r e s u l t i s n o t n e c e s s a r i l y t r u e i f t h e c o d i n e n s i o n of t h e subspace F i n t h e space E i s n o t f i n i t e : i n deed, s e t E:=l
00
and F:=co. C l e a r l y E i s a Mackey space and F 1 1 ) ) f (F,m(F,l ) ) , s i n c e o t h e r w i s e subspace o f E, b u t (F,m(E,l 1 1 1 u n i t b a l l B o f 1 would be compact i n ( 1 ,s(1 ,E)) and hence 1 ) ) thou@ n o t r e f l e x i v e , a c o n t r a d i c t i o n . Moreover, (F,m(E,l
i s a closed the closed 1 1 would be a !lackey spa-
ce can be c o n s i d e r e d as t h e c o u n t a b l e i n t e r s e c t i o n o f Plackey spaces: indeed, 1 1 s i n c e ( 1 , b ( l ,E)) i s separable, we have t h a t (F',s(F',E/F)) i s separable and 4.4.19
( b ) shows t h a t F i s t h e i n t e r s e c t i o n o f a d e c r e a s i n g sequence o f
c l o s e d f i n i t e - c o d i m e n s i o n a l subspaces o f E which a r e c l e a r l y Mackey soaces. I f S: denotes t h e f a m i l y o f a l l Mackey spaces i n 4.5.3
(ii),
and 4.5.2
t h e n we g e t t h e e x i s t e n c e o f a dense hyperplane o f a Mackey space which i s n o t a Mackey space. There i s a v e r y s i m p l e way o f c o n s t r u c t i n g e x p l i c i t axamples o f t h i s t y p e : l e t ( F , t ) F " \ F. Set E:= s p ( F U 4 x ) ) .
be a n o n - r e f l e x i v e F r g c h e t space and x i n
Since F i s dense i n (F",s(F",F')),
dual p a i r and F i s dense i n (E,m(E,F')). and hence ( F , t )
#
Thus (F,m(E,F'))
(E,F')
is a
i s n o t conplete
(F,m(E,F')).
On t h e o t h e r hand, i f G i s a Frgchet-Monte1 space, s e t E:=(G',m(G',G)) and l e t F be a dense subspace o f E. E i s a Mackey space and (F,m(G',G)) a l s o a Mackey space: indeed, s i n c e m(F,G)
i s f i n e r t h a n m(G',G)
is
on F i t i s
enough t o prove t h a t g i v e n any a b s o l u t e l y convex compact s e t A o f (G,s(G,F)) then A i s compact i n (G,s(G,G')).
Since A i s bounded i n (G,s(G,F))
n o n i c a l i n j e c t i o n J:GA-----(G,m(G,F)) c l o s e d in GAx(G.m(G,F))
t h e ca-
i s c o n t i n u o u s and hence i t s graph i s
and, a f o r t i o r i , i n GAx(G,m(G,G')),
which
c h e t space. A c c o r d i n g t o t h e c l a s s i c a l c l o s e d graph theorem, J:GA
is a Fr6-
-------
i s continuous and A i s bounded i n (G,s(G,G')). Since G i s -+(G,m(G,G')) Montel, A i s r e l a t i v e l y compact i n (G,s(G,G')). Since A i s c l o s e d i n (G, s(G,F)),
i t i s a l s o c l o s e d i n (G,s(G,G')).
Thus A i s compact i n (G,s(G,G'f).
Observe t h a t i f F i s a dense hyperplane o f (E',s(E',E))
f o r a Banach
space E, i t i s n o t n e c e s s a r i l y t r u e t h a t e v e r y compact s e t i n (E,s(E,F)) compact i n (E,s(E,E')):
i n RO,p.133
t r u c t e d such t h a t G ' and G" a r e separable and dim(G"/G) F:=G.
C l e a r l y F i s dense i n (E',s(E',E))
compact i n (E,s(E,F)).
is
a s e p a r a b l e Banach space G i s cons=
1. Set E:=G' and
and t h e c l o s e d u n i t b a l l B o f E i s
IfB i s compact i n (E,s(E,E'))
it follows that E i s
BARRELLED LOCALLY CONVEXSPACES
162
r e f l e x i v e and hence F i s r e f l e x i v e , a c o n t r a d i c t i o n . T h i s example a l l o w s us t o show t h a t t h e t o p o l o g y s(G',G") a b a s i s o f 0-nghbs which a r e c l o s e d i n (G',s(G',G)):
does n o t have
a c c o r d i n g t o K1,918.4.
( 4 ) b ) i t i s enough t o e x h i b i t a s e q u e n t i a l l y complete s e t i n (G',s(G',G)) which i s n o t s e q u e n t i a l l y complete i n (G',s(G',G")). (G',s(G',G))
Since 6 i s b a r r e l l e d ,
i s s e q u e n t i a l l y complete and so i s t h e c l o s e d u n i t b a l l B o f
G I . Since B i s bounded i n (G',s(G',G")),
B i s precompact i n i t . bforeover
(B,s(G' ,GI1)) i s m e t r i z a b l e s i n c e G" i s seDarable and 4.4.12 a D p l i e s . Should (B,s(G',G"))
be s e q u e n t i a l l y complete, i t would f o l l o w t h a t (B,s(G',G"))
were compact and t h i s would l e a d t o a c o n t r a d i c t i o n as above. Now we c h a r a c t e r i z e those F r e c h e t spaces E such t h a t e v e r y dense hyperp l a n e F o f (E',m(E',E)) Theorem 5.2.1:
i s a Mackey space.
L e t E be a Mackey space such t h a t ( E ' , s ( E ' , E ) )
complete. I f (E',s(E',E))
i s locally
i s n o t s e q u e n t i a l l y complete, t h e n t h e r e i s a
dense hyperplane F o f E which i s n o t a Mackey space. P r o o f : L e t ( v ( n):n=1,2,. s(E',E)) s(G,E))
. ) be a non-convergent Cauchy sequence i n ( E l ,
and v i t s l i m i t i n (E*,s(E*,E)).
We s e t G : = s p ( E ' u { v \ ) .
Then ( G ,
i s l o c a l l y complete because i t c o n t a i n s t h e l o c a l l y complete hyper-
p l a n e E (5.1.13).
Since (v(n)-v:n=1,2,..)
we a p p l y 5.1.11 t o g e t t h a t t h e
i s a n u l l sequence i n (G,s(G,E)),
c l o s e d a b s o l u t e l y convex h u l l B o f ( v ( n ) :
n=1,2,. . ) i s compact i n (G,s(G,E)). 1 We s h a l l see t h a t F:=v- ( 0 ) is t h e d e s i r e d hyperplane o f E. F i r s t we prove t h a t B n E ' i s compact i n (E',s(E',F)).To a r l y s(E',F)-precompact,
i t i s enough t o show t h a t i t i s c o n p l e t e . L e t
( w ( j ) : j c J ) be a Cauchy n e t i n ( B n E ' , s ( E ' , F ) ) s(F*,F).
Since B i s compact i n (G,s(G,E))
t h e n e t ( w ( j ) : j c J ) . We can w r i t e w ' XE
do t h i s , s i n c e B A E ' i s c l e -
=
and W E E *
i t s l i m i t i n (F*, there i s a cluster point W ' G B o f
av+u, w i t h a c K and U E E ' .
F we have t h a t = < x,w> = a < x,v)
For e v e r y
+ , hence u E E ' i s
t h e l i m i t of t h e n e t ( w ( j ) : j c J ) i n ( E ' , s ( E ' , F ) ) . Now we prove t h a t B"nF i s a 0-nghb i n (F,m(F,E')) which i s n o t a 0-nghb i n F. T h i s w i l l ensure t h a t F i s n o t a Mackey space. Since B n E ' i s dense i n (B,s(G,E)), i t i s a l s o dense i n (B,s(G,F)), hence B"nF, which c o i n c i d e s I f BonF i s a w i t h t h e p o l a r s e t o f B n E ' i n F, i s a 0-nghb i n (F,m(F,E')). 0-nghb i n F f o r t h e induced t o p o l o g y , t h e n B", which c o n t a i n s t h e c l o s u r e i n E o f B'nF,
i s a 0-nghb i n E. T h i s i m p l i e s t h a t t h e sequence ( v ( n ) : n = l , . )
i s equicontinuous, and hence v c E ' , a c o n t r a d i c t i o n ,
//
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163
Observe t h a t taken, f . i . , E:= ( l ~ ' , m ( l W 1 , l ~ the ) ) , method of proof of 5. 2 . 1 provides exanples of t h e s i t u a t i o n considered i n 4 . 5 . 2 ( i i ) . From now on, i f A i s a subset of a space E , we s h a l l denote by A* i t s c l o s u r e in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . To obtain a converse of 5.2.1 i n t h e case of Mackey duals of Frechet spaces we need some Lemmata. Lemna 5.2.2: Let A be a subset of a space E. Let L be a dense subspace o f ( E ' , s ( E ' , E ) ) w i t h dim(E'/L) = n , such t h a t A i s compact i n (E,s(E,L)). Then dim(sp(EuA*)/E) f n. Proof: Let ( u ( 1 ) ,...,u ( n ) ) be a co-basis o f L i n E l . Let z ( p ) , p = l , . . , n , be elements of ( E l ) * such t h a t z ( p ) vanishes on L and t z ( p ) , u ( q ) > equals 1 i f p=q and 0 i f p f q . We a r e done i f we prove t h a t A* i s included i n the span of E U ( z ( l ) , . . , z ( p ) \ . Let z be any point of A* and ( x ( j ) : j E J ) a net i n A converging t o z in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since A i s compact i n (E,s(E, L ) ) , t h e r e i s a point x i n E such t h a t ( x ( j ) : j c J ) converges t o x i n (E,s(E, L ) ) . We s e t y:=z-x- z < z - x , u ( p ) > x ( p ) . I f U C L , then - P.1 = 0 and i f p = l , ...,n , then < y , u ( p ) > = < z - x , u ( p ) > - < z - x , u ( p ) > = 0 . T h u s we have t h a t z = x + L < z - x , u ( p ) > x ( p ) . / / P'
Lemma 5.2.3: Let A be a d i s c i n a Frechet space E , such t h a t t h e d i k e n sion of s p ( E U A * ) / E i s equal t o 1. I f x belongs t o A*, t h e n t h e r e i s a sequence ( x ( n ) : n = 1 , 2 , . . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Proof: Since A* i s not included i n E , A i s not a weakly r e l a t i v e l y compact subset of E . By K1,§24.2.( 1) t h e r e i s a countable subset B of A such t h a t B * i s not included in E . Let y be a point i n B* not belonging t o E . Clearly sp(EuA*) = s p ( E U B * ) = s p ( E O{y\). Take any point x i n A*, then x= ay+z, w i t h a € K and z c E . Let F be t h e c l o s u r e i n E of sp(B U { z \ ) and G : = sp(FUCy1). Since ( F ' , s ( F ' , F ) ) i s separable, we can apply 2.5.18 t o obt a i n t h a t ( F ' ,s( F ' , G ) ) i s separable, hence ( B * , s ( G , F ' ) ) i s m t r i z a b l e . Thus t h e r e i s a sequence ( y ( n ) : n = 1 , 2 , . . ) i n B converging t o y i n (G,s(G,F')). Now s e t t i n g x ( n ) : = ay(n) + z , n = 1 , 2 , ..., we obtain t h e desired sequence i n E converging t o x i n ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . , / Theorem 5.2.4: Let E be a Frechet space. Every hyperplane of (E',m(E',E)) i s a Mackey space i f and only i f ( E , s ( E , E ' ) ) is s e q u e n t i a l l y complete.
BARRELLED LOCALLY CONVEXSPACES
164
Proof: I f ( E , s ( E , E ' ) ) i s not s e q u e n t i a l l y complete, we apply 5.2.1 t o obtain a hyperplane of E' which i s not a Mackey space. Conversely, suppose t h a t ( E , s ( E , E ' ) ) i s s e q u e n t i a l l y complete. Let F be a hyperplane of E ' . If F i s closed i n ( E ' , m ( E ' , E ) ) , then i t i s c l e a r l y a Mackey space. If F i s dense, i t i s enough t o show t h a t every compact d i s c in ( E , s ( E , F ) ) i s a l s o compact i n ( E , s ( E , E ' ) ) . To prove t h i s , l e t 63 be the c l a s s of a l l compact d i s c s in ( E , s ( E , F ) ) , and l e t t be t h e l o c a l l y convex topology on E such t h a t ( E , t ) i s t h e inductive l i m i t of t h e family of Banach spaces (Eg:BEB ) . Since t i s f i n e r than m ( E , F ) , t h e i d e n t i t y mapping I:(E,t)----(E,m(E,F)) i s continuous. On t h e other hand, m ( E , F ) i s coarser than t h e i n i t i a l topology of t h e FrGchet space E , hence I : ( E , t ) - - - + E has closed graph i n ( E , t ) * E . By 1.2.19 we obtain t h a t I is continuous, t h e r e f o r e B i s a bounded subset of E
f o r every B C 8 . We apply 5.2.2 t o obtain t h a t d i m ( s p ( E U B * ) / E ) 5 1 f o r every BE^ . According t o 5.2.3, i f xcB* t h e r e i s a sequence ( x ( n ) : n = l , . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since ( E , s ( E , E ' ) ) i s sequent i a l l y complete, x belongs t o E . Therefore B* i s included i n E,and t h u s every B E 63 i s compact in (E,s( E Y E ' ) ) .
//
5.3 Notes and renarks. Local convergence was considered by M A C K E Y , ( I ) and GROTHENDIECK, see G , Ch.3,4, exerc. 5 where some p a r t s of 5.1.11 appear. A space i s p-complete (polar-semireflexive o r topologically complete) i f every precompact subset of E i s r e l a t i v e l y compact ( s e e K I , p.311 and DAY, (21, p.53 r e s p e c t i v e l y ) . A space E i s r - c o n p l e t e i f every sequence s a t i s f y i n g the Cauchy condition (2.2.6) i s s u m b l e . One has t h e following chain of i m l i c a t i o n s complete+quasi-conpleteep-conplete plete.
+ sq-cormlete 3 t - c o m l e t e
=+local corr-
( 1 1 , s ( l 1 , 1 ~ ) )i s a non-p-conplete s e q u e n t i a l l y corrplete space. If E i s the r e f l e x i v e Banach space constructed by JAMES,( 1) w i t h E ' , E l ' , . . . . separab l e , i t can be shown t h a t ( E ' , s ( E ' , E " ) ) i s a non-sequentiallv complete 2conplete space ( s e e DIEROLF,( 2 ) ,p.25). The sDace ( co.s( c Q , l 1 ) ) i s a non- 'Lcomplete l o c a l l y conplete space ( j u s t consider t h e canonical u n i t v e c t o r s ! ) . Weakly 2 - c o m p l e t e spaces a r e 2 - c o n p l e t e and the weakly 2 - c o n p l e t e Banach spaces a r e p r e c i s e l y those which do n o t contain co ( s e e BESSAGA,PELCZYNSKI,(4)). Moreover, i f E i s a l o c a l l y complete space such t h a t every continuous l i n e a r operator T:C( K ) - ( E , s ( E , E ' ) ) i s compact, f o r every compact space K , then E i s weakly L-conplete ( t h a t i s , ( E , s ( E , E ' ) ) i s Z - c o m l e t e ) , see THOWAS,( 1). The c h a r a c t e r i z a t i o n of local completeness a s presented in 5.1.11 i s d u e t o DIEROLF,(5). Let us observe t h a t t h i s result f a i l s t o be true in general in t h e non-locally convex s e t t i n g ( s e e DIEROLF,(Z): s e t p : = 1 / 2 and consider
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the F-space ( l P , q ) . S e t u(n):=e(n)/f?? , e ( n ) being t h e n - t h canonical u n i t vector. ( u ( n ) : n = 1 , 2 , . . ) is a n u l l sequence i n ( P , q ) and v ( n ) : = ( n - I ) . T u ( i ) belong t o acx(u(n):n=1,2,..). Since q(v(n))’nl/I, one has t h a t acx(u(n):n= l Y Z y .. ) i s not even bounded). 5.1.15 and 5.1.16 a r e taken from MAROUINA,PEREZ CARRERAS ,( 4 ) . 5.1.26 can be seen i n DIEROLFy(5). The M.c.c. and s.M.c. were introduced i n GROTHENOIECK ,(2) where 5.1.31 i s a l s o remarked. 5.1.32 i s due t o WEBB,(4). In the context of MAHWALD-type closed graph t h e o r e m one has t h e f o l l o w i n g remrkable result 5.3.1: (VALDIVIA,(43)) The weak dual of a space E i s l o c a l l y complete i f and only i f every l i n e a r mapDing f : E - + 1 2 with closed graph i s weakly c o n t i nuous. All t h e r e s u l t s which appear in 5.2 a r e taken from VALDIVIA, (13) where many o t h e r r e s u l t s on t h e s t a b i l i t y of Mackey spaces by subspaces a r e i n cluded. W e l i s t a few of them. 5.3.2: Every dense subspace of a r e f l e x i v e (LF)-space i s a Mackey soace. 5.3.3: Every dense subspace of a q u a s i b a r r e l l e d (DF)-space is a Mackey soace. 5.3.4: Every subspace of a countable product of nuclear (DFf-sDaces is a R i G y space.
CHAPTER FIVE LOCAL COMPLETENESS
5 . 1 D e f i n i t i o n s and c h a r a c t e r i z a t i o n s . D e f i n i t i o n 5.1.1:
L e t E be a space. A sequence (x(n):n=1,2,..)
in E is
s a i d t o be l o c a l l y convergent o r Mackey convergent t o an element x o f E if t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o x i n
b. A
se-
quence i s l o c a l l y n u l l i f i t i s l o c a l l y convergent t o t h e o r i g i n . A sequence i s c a l l e d l o c a l l y Cauchy o r Mackey Cauchy i f i t i s a Cauchy sequence i n
5
f o r a c e r t a i n d i s c B i n E. Lemm 5.1.2:
L e t E be a n e t r i z a b l e space and (x(n):n=1,2,..)
a n u l l se-
quence i n E, t h e n t h e r e i s an i n c r e a s i n g unbounded sequence o f p o s i t i v e r e s l numbers (a(n):n=1,2,..)
such t h a t t h e sequence (a(n)x(n):n=l,Z,..)
con-
verges t o t h e o r i g i n . P r o o f : L e t (Uk:k=1,2,..)
be a d e c r e a s i n g b a s i s o f a b s o l u t e l y convex
0-nghbs i n E. Since (x(n):n=1,2,..)
converges t o t h e o r i g i n , we can f i n d
an i n c r e a s i n g sequence (n(k):k=1,2,. . ) o f p o s i t i v e i n t e g e r s such t h a t 1 x ( n ) € k- Uk, n 2 n ( k ) , k=1,2, ... We s e t a ( n ) : = l i f l L n < n ( l ) and a(n):=k i f
...
n < n ( k + l ) , k=1,2, Then a ( n ) x ( n ) e Uk f o r e v e r y n’n(k), k=1,2, n(k) Thus t h e sequence (a(n)x(n):n=1,2,..) converges t o t h e o r i g i n i n E.
...
//
P r o p o s i t i o n 5.1.3:
(i) A sequence (x(n):n=1,2,..)
converges t o x i f and o n l y i f (x(n)-x:n=l,Z,..) ( i f ) A sequence (x(n):n=1,2,..)
i n a space E l o c a l l y
i s locally null.
i n a space E i s l o c a l l y n u l l i f and o n l y if
t h e r e i s an i n c r e a s i n g unbounded sequence ( a ( n):n=1,2,. numbers such t h a t (a(n)x(n):n=1,2,..)
.) o f p o s i t i v e r e a l
converges t o t h e o r i g i n i n E.
P r o o f : (i) i s t r i v i a l . To prove ( i i ) , i f (a(n):n=1,2,..)
i s an i n c r e a s -
i n g unbounded sequence o f p o s i t i v e r e a l numbers such t h a t (a( n ) x ( n ) :n=I,. . ) converges t o t h e o r i g i n , t h e s e t B : = ZEx(a(n)x(n):n=1,2,..)
i s a closed
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152
d i s c i n E such t h a t the sequence ( x ( n ) : n = l , Z , . . ) converges t o t h e o r i g i n in Conversely, i f ( x ( n ) : n = 1 , 2 , . . ) i s l o c a l l y n u l l , t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o the o r i g i n in EB. Applying 5.1.2 t h e r e i s an incresing unbounded sequence of p o s i t i v e real numbers ( a ( n ) : n = 1 , 2 , . . ) such t h a t ( a ( n ) x ( n ) : n = l , Z , . . ) converges t o t h e o r i q i n i n and
k.
hence i n E .
/I
Proposition 5.1.4:A sequence in a metrizable space i s convergent i f and only i f i t i s l o c a l l y convergent. Proof: I t i s enough t o apply 5.1.2 and 5 . 1 . 3 ( i i ) .
//
Definition 5.1.5: A space is l o c a l l y conplete i f every l o c a l l y Cauchy sequence i s 1ocal l y convergent. Proposition 5.1.6: Let E be a space. The following conditions a r e equiVal e n t : ( i ) E i s l o c a l l y complete. ( i i ) Every closed d i s c in E i s a Banach d i s c . ( i i i ) I f u i s a topology of t h e dual pair(E,E') and B i s a d i s c in E , then every Cauchy sequence i n i s convergent in ( E , u ) . ( i v ) Every bounded subset of E i s included i n a Banach d i s c . Proof: ( i ) - + ( i i ) . Let B be a closed d i s c in E and ( x ( n ) : n = 1 , 2 , . . ) a Cauchy sequence i n g. By ( i ) t h e r e i s x in E such t h a t t h e sequence converges l o c a l l y t o x and hence converges i n E . Since ( x ( n ) : n = 1 , 2 , . . ) is bounded in E B , t h e r e i s a > O such t h a t x ( n ) E a B , n = 1 , 2 , ... Therefore xEaB, B being closed in E . Moreover f o r every b>O t h e r e i s a p o s i t i v e i n t e g e r p such t h a t x ( n ) - x ( m ) ~ b Bf o r every n , m 2 p. L e t t i n g m t o i n f i n i t y we have
5
that x(n)-xEbB i f n
p . T h u s t h e sequence converges t o x i n
%
and EB i s
complete. ( i i ) - - b ( i i i ) . Let u be a topology of the dual p a i r ( E , E ' ) and A a d i s c in E . Set B f o r the c l o s u r e of A i n E. I f ( x ( n ) : n = l , Z , . . ) i s a Cauchy sequenc e i n E A y then i t i s a l s o a Cauchy sequence i n t h e Banach space E B , hence ( x ( n ) : n = 1 , 2 , . . ) converges i n Since the canonical i n j e c t i o n from EB i n t o ( E , u ) i s continuous, i t follows t h a t t h e sequence converges i n ( E , u ) . ( i i i ) - - b ( i ) . Let ( x ( n ) : n = l , Z , . . ) be a Cauchy sequence i n f o r a certain d i s c B . By ( i i i ) t h e r e i s x i n E such t h a t the sequence converges t o x i n ( E , s ( E , E ' ) ) . On t h e o t h e r hand f o r every b > O t h e r e i s a p o s i t i v e i n t e g e r
k.
5
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153
p such t h a t i f n , m 2 p, then x(n)-x( m)E tRc bC, where C i s the closure of B i n E . Letting m t o i n f i n i t y we have t h a t x(n)-xebC i f n 5 p , and t h u s the sequence ( x ( n ) : n = 1 , 2 , . . ) converges l o c a l l y t o x. ( i i ) - + ( i v ) and ( i v ) - - b ( i ) a r e t r i v i a l .
//
Corollary 5.1.7:
If a space E is l o c a l l y complete, t h e n i t i s l o c a l l y
complete f o r every topology of the dual p a i r ( E , E ' ) . Corollary 5.1.8: If a space i s sequentially complete, then i t is local l y conplete. Proof: Every closed d i s c i n a sequentially complete space i s a Banach d i s c by 3.2.5.
//
Corollary 5.1.9: A netrizable space i s l o c a l l y complete i f and only i f i t i s complete. Proof: Apply 5.1.8 and 5.1.4.
//
Corollary 5.1.10: Every barrel i n a l o c a l l y complete space i s bornivorous. I n p a r t i c u l a r a l o c a l l y complete space i s barrelled i f and only i f i t i s quasi barrel led. Proof: By 3.2.7 b a r r e l s absorb Banach d i s c s .
//
Theorem 5.1.11: Let ( E , t ) be a space. The following conditions a r e equivalent : ( i j ( E , t ) i s l o c a l l y complete. (ii) The closed absolutely convex hull of every l o c a l l y null sequence i n ( E , t ) i s compact. ( i i i ) The closed absolutely convex hull of a null sequence in ( E , s ( E , E ' ) ) i s compact i n ( E , s ( E , E ' ) ) . ( i v ) The closed absolutely convex hull of a null sequence i n ( E , t ) i s compact. Proof: ( i ) - - - ( i i i ) . Let ( x ( n ) : n = 1 , 2 , . . ) be a null sequence i n E endowed w i t h i t s weak topology. Let B be i t s closed absolutely convex h u l l . Since E i s l o c a l l y conplete, EB i s a Banach space, hence we apply 3.2.12 t o obt a i n t h a t B i s compact i n ( E , s ( E , E ' ) ) . ( i i i ) - + ( i v ) . According t o ( i i i ) the closed absolutely convex hull of a
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154
n u l l sequence i n ( E , t )
i s s(E,E')-compact
and t-precompact,
nence K1§18,4,
( 4 ) i m p l i e s t h a t i t i s a l s o compact i n ( E , t ) . ( i v ) - + ( i i ) i s t r i v i a l s i n c e e v e r y l o c a l l y n u l l sequence i n E i s n u l l i n (E,t). (ii)--(i).
5 . We
a Cauchy sequence i n
L e t B be a d i s c i n E and (x(n):n=l,Z,..)
o f positive ink and we s e t y ( k ) : = 2 ( x ( n k t 1 ) - d n k ) ) .
s e l e c t a s t r i c t l y i n c r e a s i n g sequence (nk:k=1,2,..)
t e g e r s such t h a t x( nktl)-x(nk)E C l e a r l y (y(k):k=1,2,..)
2-'%
converges t o t h e o r i g i n i n
5
c l o s e d a b s o l u t e l y convex h u l l A i s compact i n ( E , t ) . '
k z ( p ) : = 5 2 - y ( k ) , p=1,2,..,
hence, by ( i i ) , i t s Now t h e sequence
i s a Cauchy sequence i n ( E , t )
hence t h e r e i s an element x i n E such t h a t (z(p):p=1,2,..) i n (E,t).
c o n t a i n e d i n A, converqes t o x
) - x ( n ), t h e sequence (x(n):n=l,Z,..) converP+l P Thus, a o p l y i n g 5.1.6 (iii), t h e space ( E , t ) i s
Since z ( p ) = x ( n
ges t o x(nl)+x
i n (E,t).
l o c a l l y complete.
Example 5.1.12:
/I L o c a l l y c o n p l e t e spaces which a r e n o t s e q u e n t i a l l y com-
p l e t e . Every F r e c h e t space endowed w i t h i t s weak t o p o l o g y i s l o c a l l y c o r n p l e t e by 5.1.7. I n p a r t i c u l a r ( c o , s ( c o y l 1) ) i s l o c a l l y complete. I f e ( n ) i t i s easy t o see t h a t denotes t h e n - t h c a n o n i c a l u n i t v e c t o r i n c 10, x ( n ) : = e( 1)+...+ e ( n ) , n=l,Z,..,is a s ( c o y l )-Cauchy sequence i n co which
does n o t converge, hence (co,s(co,l
1
) ) i s n o t s e q u e n t i a l l y complete. More
g e n e r a l l y , if E i s a n o n - r e f l e x i v e F r e c h e t space, whose s t r o n g dual i s separable, t h e n (E,s( E Y E ' ) ) i s l o c a l l y complete b u t n o t s e q u e n t i a l l y c o m l e t e . Indeed, t a k e a p o i n t
ZE
E l ' \ E. There i s a bounded subset B of E such
t h a t z belongs t o t h e c l o s u r e C o f B i n (E",s(E",E')). i s separable, (C,s(E",E'))
n=1,2,..)
i n B c o n v e r g i n g t o z i n (E",s(E",E')).
a Cauchy sequence i n (E,s(E,E')) P r o p o s i t i o n 5.1.13:
t e , t h e n E i s l o c a l l y complete. P r o o f : L e t B be a c l o s e d d i s c i n E . The space space and a c l o s e d hyperplane o f
D e f i n i t i o n 5.1.14:
Thus ( x ( n ) : n = l , 2 , . . . ) i s
which does n o t converge.
L e t F be a hyperplane o f
complete spaces and hence
Since (E',b(E',E))
i s t r e t r i z a b l e , hence t h e r e i s a sequence ( x ( n ) :
5 . Thus
E.
I f F i s l o c a l l y comple-
hnF=
FBnF i s a Banach
EB i s complete as a p r o d u c t of
E i s l o c a l l y conplete.
//
L e t E be a space and A a non-void subset o f E . A
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155
point x i s a local l i m i t point of A i f t h e r e is a sequence i n A l o c a l l y convergent t o x. We say t h a t A i s l o c a l l y closed i f every local limit point of A belongs t o A. A s e t B i s l o c a l l y dense i n A i f every point of A i s a local l i m i t point of B . We assume t h a t t h e void set i s l o c a l l y closed. Exanple 5.1.15: Let ( E i : i E I ) be an i n f i n i t e family of spaces and E i t s product. Let x = ( x ( i ) : i ~I ) be an element of Eo. There i s a sequence J:= ( i ( n ) : n = l , Z , . . ) i n I such t h a t x ( i ) = 0 i f i e I ' J . We s e t x n : = ( x n ( i ) : i E I ) defined by s ( i ) = O i f i c I \ J o r i = i ( m ) , m z n , and x,(i(m)) = x ( i ( m ) ) i f m=l, n . We s h a l l s e e t h a t t h e r e i s an absolutely convex compact subset C of Eo such t h a t x ( n ) converges t o x i n ( E o ) C . We take B i :={O\ i f i c I \ J
...,
and B : = a c x ( ( p x ( i ( p ) ) ) ) , p = 1 , 2 , ... Clearly t h e s e t B : = TI ( B i : i € I ) i s i ( P) a compact subset of Eo. On the o t h e r hand i f C is t h e closed a b s o l u t e l y convex hull of ( n ( x n - x ) : n = 1 , 2 , . . ) , i t i s obvious t h a t CcB and x - x n c ( l / n ) C f o r n = 1 , 2 , . . , from where i t follows t h a t C i s a compact absolutely convex subset of Eo and t h a t xn converges t o x i n ( E o I C . i n p a r t i c u l a r we obtain the following Proposition 5.1.16:
If ( E i : i €
I ) i s a non-void family of spaces and E
i s i t s product, then e ( E i : i e I ) is l o c a l l y dense in Eo. Proposition 5.1.17: The i n t e r s e c t i o n of l o c a l l y closed sets i s l o c a l l y closed. Definition 5.1.18: The local closure of a subset A of a space E i s t h e i n t e r s e c t i o n of a l l t h e l o c a l l y closed subsets of E containing A . By 5.1. 17 the local closure of A i s l o c a l l y closed and contains a l l t h e local l i m i t points of A . Definition 5.1.19: A subset A of a space E i s s a i d t o be lo c a ll y comp l e t e i f every local Cauchy sequence i n A converges l o c a l l y t o a point of
A.
Proposition 5.1.20: ( i ) Every l o c a l l y complete subset of a space E i s l o c a l l y closed. ( i i ) Every l o c a l l y closed subset of a l o c a l l y complete space E i s l o c a l l y compl e t e .
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156
I f E i s a space, t h e l o c a l c o m p l e t i o n o f E i s d e f i -
D e f i n i t i o n 5.1.21:
ned as t h e l o c a l c l o s u r e o f E i n i t s completion. I t i s denoted b y ? . 1.20,
? coincides
By 5.
w i t h t h e i n t e r s e c t i o n o f a l l t h e l o c a l l y complete sub-
spaces o f t h e c o m p l e t i o n o f E c o n t a i n i n g E . Observation 5.1.22:
By 5.1.4
e v e r y m t r i z a b l e space i s l o c a l l y dense i n
i t s completion, hence t h e l o c a l Completion o f a m e t r i z a b l e space c o i n c i d e s w i t h i t s completion. Lemma 5.1.23:
L e t E and F be two spaces and f:E-----*F
a continuous li-
near mapping. Then
( i )I f (x(n):n=1,2,..)
converges l o c a l l y t o x, t h e n (f(x(n)):n=1,2,
... )
converges l o c a l l y t o f( x).
(ii)If A i s a l o c a l l y c l o s e d subset of F, t h e n f - l ( A ) i s a l o c a l l y c l o s e d subset o f E. P r o o f : ( i ) i s obvious. To prove ( i i ) , l e t (x(n):n=1,2,..) be a sequence 1 i n f- ( A ) l o c a l l y convergent t o x i n E. By ( i ) t h e sequence ( f ( x ( n ) ) : n = l , Z,..)
i s l o c a l l y convergent t o f ( x ) . Since A i s l o c a l l y c l o s e d , f ( x ) be-
1
l o n g s t o A, t h u s x belongs t o f - ( A )
P r o p o s i t i o n 5.1.24:
-11
L e t f:E------F
be a c o n t i n u o u s l i n e a r mapping. I f
A i s a subset o f E, t h e n t h e image o f t h e l o c a l c l o s u r e o f A by f i s i n cluded i n t h e l o c a l c l o s u r e of f ( A ) . P r o o f : Take any p o i n t x i n t h e l o c a l c l o s u r e o f A and B any l o c a l l y c l o s e d subset o f F c o n t a i n i n g f ( A ) . T h e r e f o r e A i s i n c l u d e d i n t h e l o c a l l y 1 o f E. Then x c f - ( B ) , hence f ( x ) a B and conseauently c l o s e d subset f - ' ( B ) f ( x ) belongs t o t h e l o c a l c l o s u r e o f f ( A ) . / /
P r o p o s i t i o n 5.1.25: mapping f : E - - - - -
L e t E and F be spaces. Given a continuous l i n e a r --&,
F, t h e r e i s a unique c o n t i n u o u s l i n e a r mapping f : E - - - - - F
N
whose r e s t r i c t i o n t o E c o i n c i d e s w i t h f .
-
P r o o f : The uniqueness i s t r i v i a l . On t h e o t h e r hand, g i v e n f t h e r e i s a f i A
continuous l i n e a r e x t e n s i o n t o t h e completions, f : E - - - - + we show t h a t ?(:) f(E)
F. We a r e done i f
i s i n c l u d e d i n ? . To see t h i s , observe t h a t E i s t h e l o -
cal closure o f E i n A
A
?.
By 5.1.24,
;(?)
i s included i n the l o c a l closure o f N
i n F which i s c l e a r l y a subset o f F.
//
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157
Corollary 5.1.26: Let E be a metrizable space and F a l o c a l l y complete space. Given a continuous l i n e a r mapping f:E-----cF, t h e r e i s a continuous l i n e a r extension ? t o t h e completion of E w i t h values i n F. Proof: Apply 5.1.22 and 5.1.25.
//
Convergence and local convergence coincide i n a metrizable space. Much more can be s a i d . Theorem 5.1.27: Let E be a metrizable space. Then ( i ) ( 1) For every sequence of bounded s e t s (An:n=1,2,. .) there a r e c( n ) > 0, n = 1 , 2 , . . , such t h a t U ( c ( n ) A n : n = 1 , 2 , . . ) i s a bounded subset of E. ( 2 ) For every sequence of bounded s e t s ( A n : n = 1 , 2 , . . ) t h e r e is a closed d i s c A such t h a t each An i s bounded i n EA. ( i i ) For every bounded subset A of E t h e r e i s a closed d i s c 6 such t h a t A i s included i n 6 and t h e topologies induced on A by E and coincide.
5
Proof: Let (U :n=1,2,..) be a decreasing b a s i s of closed absolutely conn vex 0-nghbs i n E . ( i ) ( l ) For every p o s i t i v e i n t e g e r n we determine c ( n ) > O such t h a t c(n)An i s included i n U n . One e a s i l y sees t h a t U(c(n)An:n=1,2,..) i s bounded i n E . ( i ) ( 2 ) Proceeding as i n the former proof i t i s enough t o take a s A t h e c l o sed absolutely convex hull of U ( c ( n ) A n : n = 1 , 2 , . . ) . ( i i ) We can suppose,without loss of g e n e r a l i t y , t h a t A i s absolutely convex. F i r s t we f i n d a closed d i s c 6 i n E containing A such t h a t f o r every a z O t h e r e i s a p o s i t i v e i n t e g e r n w i t h A n U n c a B . Given A t h e r e i s c ( i ) > O such We determine b ( i ) 4 c ( i ) such t h a t t h e seauence that ACc(i)Ui, i=1,2, ( c ( i ) b ( i ) - ' : i = l , Z , . . ) converges t o zero. We set B : = n ( b ( i ) U i : i = l , 2,...). Clearly B i s a closed d i s c i n E containing A. Given a > O there i s a p o s i t i ve i n t e g e r j such t h a t i f i 2 j , t h e n c ( i ) < a b ( i ) , and t h e r e f o r e ACab(i)Ui i f i 2 j . On t h e o t h e r hand n ( a b ( i ) U i : i = l , . . , j - l ) i s a 0-nghb in E , hence
...
t h e r e is a p o s i t i v e i n t e g e r n with Unc ~ ( a b ( i ) U i : i = l Y . . . j - l ) ,from this i t follows t h a t AnUn C n ( a b ( i ) U i : i = 1 , 2 , . . ) = aB. Now t o prove t h a t E and EB induce t h e same topology on A i t i s enough t o show t h a t both induced topol o g i e s have the same b a s i s o f 0-nghbs i n A, by RR,ch6,1,Lemnal. Since t h e topology of EB i s f i n e r than t h e topology of E, t h e conclusion follows from our construction of B
-//
-Corollary
5.1.28:
I f E i s a metrizable space and A i s a preconpact ( r e s p .
BARRELLED LOCALLY CONVEXSPACES
158
compact) subset o f E, t h e n t h e r e i s a c l o s e d d i s c B c o n t a i n i n g A such t h a t
A i s precompact ( r e s p . compact) i n ER. Proof: R e c a l l t h a t n o t o n l y t h e t o p o l o g i e s induced by E and
5
coincide
on A b u t a l s o t h e u n i f o r m i t i e s , by K1,§28.6.(3).//
Theorem 5.1.27
and P r o p o s i t i o n 5.1.2
suggest t h e f o l l o w i n g d e f i n i t i o n s
due t o GROTHENDIECK.
A space E i s s a i d t o s a t i s f y t h e Mackey convergence
D e f i n i t i o n 5.1.29: c o n d i t i o n (M.c.c.)
i f e v e r y n u l l sequence i n
E i s l o c a l l y n u l l . A space E
i s s a i d t o s a t i s f y t h e s t r i c t Mackey c o n d i t i o n ( s . M . c . )
i f f o r e v e r y boun-
ded subset A o f E t h e r e i s a c l o s e d d i s c B such t h a t t h e t o p o l o g i e s induced on A by
E
and
coincide.
Observation 5.1.30: s.M.c.
( i ) By 5.1.27 e v e r y m e t r i z a b l e space s a t i s f i e s t h e
(ii)I f a space s a t i s f i e s t h e s.M.c., ( i ) I f (E,t)
P r o p o s i t i o n 5.1.31: t h e M.c.c.)
t h e n i t s a t i s f i e s t h e M.c.c.
i s a space s a t i s f y i n g t h e s.M.c.(resp.
and F i s a subspace o f E, t h e n ( F , t )
s a t i s f i e s t h e s.M.c.
(resp.
t h e M.c.c.). ( i i ) I f ((En,tn):n=l,2,..)
. ) and
t h e n Ti(( En,tn):n=1,2,.
( r e s p . t h e M.c.c.), t i s f y t h e s.M.c.
i s a sequence o f spaces s c t i s f y i n g t h e s.M.c.
.. )
e(( En,tn):n=1,2,.
sa-
( r e s p . t h e M.c.c.).
P r o o f : (i)l. Suppose
E has t h e M.c.c.
and l e t (x(n):n=1,2,..)
be a n u l l
There i s a c l o s e d d i s c B i n E such t h a t t h e sequence
sequence i n ( F , t ) .
converges t o t h e o r i g i n i n EB, hence C:=Br\F i s a c l o s e d d i s c i n F such converges t o t h e o r i g i n i n Fc. 2 . Now we suppose t h a t
t h a t (x(n):n=1,2,..)
E has t h e s.M.c.
and t a k e a d i s c A i n F. There i s a c l o s e d d i s c B i n E con-
t a i n i n g A such t h a t , f o r e v e r y a,O,
t h e r e i s a 0-nghb V i n E w i t h V n A c a B .
Taking C:=BAF we have t h a t C i s a c l o s e d d i s c i n F c o n t a i n i n g A such t h a t f o r e v e r y a > O t h e r e i s a 0-nghb V ( \ F i n F w i t h V n F A A C a B n F ( i i ) l . We s e t G:= TT ( ( E n , t n ) :n=1,2,. n i c a l p r o j e c t i o n . L e t (x(k):k=1,2,..) (x(k,n):n=1,2,..),
k=1,2
,...
a c l o s e d d i s c B n i n (En,tn) origin i n (E )
. We
. ) and w r i t e p,:G-----E
aC.
m be a n u l l sequence i n G w i t h ~ ( k =)
I f each ( E n , t n )
s a t i s f i e s t h e M.c.c.
such t h a t (x(k,n):n=1,2,..)
s e t B:=TT(nBn:n=1,2,..),
Bn prove t h a t (x(k):k=1,2,..)
=
f o r t h e canothere i s
converges t o t h e
which i s a d i s c i n G. We
converges t o t h e o r i g i n i n GB. Given a > O t h e r e
CHAPTER 5
159
i s a p o s i t i v e i n t e g e r m with na > 1 i f n , m y and t h e r e i s a p o s i t i v e i n t e g e r s such t h a t x ( k , n ) c aBn i f n = l , . . , m and k 5 s . On the o t h e r hand i f k s and n > m , we have t h a t x ( k , n ) E B n , hence x ( k , n ) E n - l p n ( B ) c p n ( a B ) . Thus i f k s we have t h a t x( k ) C aB. 2. Suppose t h a t each ( E n y t n ) s a t i s f i e s t h e s.M.c. and l e t A be a d i s c i n G. Clearly A c T i ( p n ( A ) : n = 1 , 2 , . . ) . For every p o s i t i v e i n t e g e r n t h e r e i s a closed d i s c B n i n ( E n y t n ) such t h a t p n ( A ) C B n and t h e topologies induced by ( E n , t n ) and t h e normed space generated by B n coincide on p n ( A ) . We s e t B : = V ( n B n : n = 1 , 2 , . . ) , which i s a closed d i s c i n G. Given a > O , t h e r e i s a pos i t i v e i n t e g e r m such t h a t na.1 i f n>m. Now i f n = l , , , , m t h e r e i s a closed absolutely convex 0-nghb Vn i n ( E n y t n ) such t h a t V n ~ p n ( A ) c a B nW . e set V:= 77 ( V n : n = l , . . , m ) x l T ( E n : n = m t l , m + 2 , . . . ) . Clearly V i s a 0-nghb i n G and
VnAcaB. T h u s
G s a t i s f i e s t h e s.M.c.
3. Observe t h a t , according t o ( i ) , ( i i ) 1 and 2 , i f E l ,
...,E P
satisfy the ...,p ) s a t i s -
s.M.c. ( r e s p . t h e M . c . c . ) , then n ( E i : i = l ,... , p ) = Q ( E . : i = l , 1 f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) . 4. I f ( x ( k ) : k = l . Z , . . ) i s a sequence i n @ ( ( E , t n ) : n = 1 , 2 , . . ) converging t o n the o r i g i n , w i t h x ( k ) = ( d k , n ) : n = l , Z , . . ) , k=1,2,.., then there is a p o s i t i v e i n t e g e r m w i t h x(k,n)=O i f n > m and k = 1 , 2 , . . and ( x ( k ) : k = l , Z , . . ) converges to the origin i n €B((Enytn):n=l m). On t h e o t h e r handyif A i s a bounded subset of @ ( ( E n , t n ) : n = l , 2 ,...), t h e n t h e r e a r e a p o s i t i v e i n t e g e r m and bounded subsets A of ( E n , t ) , n = l , . . , m , such t h a t A i s included in
,...,
n
n
@(An:n=1,2,..). Now we can apply 3 t o obtain t h a t i f ( E n y t n ) s a t i s f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) , n = 1 , 2 , . . , then e((E, t ) : n = l , Z , . . ) a l s o s a t i s n n f i e s the s.M.c. ( r e s p . the M . c . c . ) .
//
Example 5.1.32: Uncountable products of spaces s a t i s f y i n g t h e s.M.c. nay f a i l the M.c.c. Let I be t h e s e t of a l l increasing unbounded sequences of p o s i t i v e real numbers. We take E:= T T ( R i : i c I ) , where each Ri i s a copy o f t h e r e a l s . Each i E I i s a sequence ( i ( n ) : n = l , 2 , . . ) i n R . For every p o s i t i v e i n t e g e r n we s e t x ( n ) : = ( x ( n , i ) : i € I ) € E defined by d n , i ) : = i ( n ) - ' , f o r every i c I . Clearly t h e sequence ( x ( n ) : n = l , Z , . . ) converges t o the o r i g i n i n E , b u t i t i s not l o c a l l y convergent, f o r i f i t were t h e r e would be an e l e ment i c 1 such t h a t ( i ( n ) x ( n ) : n = 1 , 2 , . . ) would converge t o the o r i g i n i n E ( 5 . 1 . 2 ) . Then t h e limit of t h e sequence ( i ( n ) x ( n , i ) : n = l , Z , . . ) i n R would be zero, which i s inpossible s i n c e i t converges t o 1.
160
BARRELLED LOCAL L Y CON VEX SPACES
Now we consider the r e l a t i o n of local conpleteness and barrelledness properties. Proposition 5.1.33: Let ( E , t ) be a space such t h a t t = d E . E ' ) . The spac e ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f every null sequence i n E-equicontinuou s . ( E ' ,s( E ' , E ) ) i s Proof: Applying 5.1.11, ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f t h e closed absolutely convex hull of every null sequence i n ( E ' , s ( E ' , E ) ) i s compact, and this i s t r u e i f and only i f every null sequence i n ( E ' , s ( E ' , E ) ) i s an ( E , m ( EYE'))-equicontinuoum.
//
Proposition 5.1.34: If E i s a space whose weak dual is l o c a l l y complete, then every barrel in E i s bornivorous. Proof: Suppose t h a t T i s a barrel in E and B a bounded subset of E not absorbed by T . For every p o s i t i v e i n t e g e r n t h e r e i s dn)eB such t h a t x ( n ) # n 2T. Applying Hahn-Banach's Theorem, we can obtain a sequence ( u ( n ) : n = 1 , 2 , . . ) i n E' such t h a t < x ( n ) , u ( n ) > = n and u ( n ) c ( n T ) : n = 1 , 2 , . . Clearly ( u ( n ) : n = 1 , 2 , . . ) converges t o t h e o r i g i n i n ( E ' , s ( E ' , E ) ) . Since ( E ' , s ( E ' , E ) ) i s l o c a l l y complete, t h e closed a b s o l u t e l y convex hull C of ( u ( n ) : n = 1 , 2 , . . ) i s compact i n ( E ' , s ( E ' , E ) ) , t h e r e f o r e i t s polar V:=C" i n E i s a 0-nghb i n (E,m(E,E')). As B i s a bounded subset of E , t h e r e i s a > O such t h a t B c a V = aC; hence l < x ( n ) , u ( n ) > [ 6 a , n = 1 , 2 , . . , which i s a c o n t r a d i c t i o n .
//
Corollary 5.1.35: A space E i s b a r r e l l e d i f and only if E i s quasibarrel l e d and ( E ' , s ( E' ,E))
i s l o c a l l y complete.
We s h a l l s e e l a t e r ( 8 . 2 ) t h a t t h e local completeness of t h e weak dual i s , in a sense, t h e weakest barrelledness condition.
5.2 S t a b i l i t y of Mackey spaces.
Now we t u r n our a t t e n t i o n t o t h e problem of when a subspace o f a Mackey space i s i t s e l f a Mackey space. More p r e c i s e l y , i f ( E , F ) i s a dual p a i r , t a topology on E compatible w i t h t h e dual p a i r , G a subspace o f E and i f ( E , t ) i s a Mackey space, i . e . , t=m(E,E'), when does ( G , t ) coincide with (G,m(G,(G,t)
')I?
CHAPTER 5
161
F i r s t observe t h a t , s i n c e t h e p r o p e r t y o f b e i n g a Mackey space is p r e served by separated q u o t i e n t s ( s e e K1,92.2.(
3 ) ) , complemented subspaces of
Mackey spaces a r e Mackey spaces. I n p a r t i c u l a r , c l o s e d f i n i t e - c o d i w n s i o n a l subspaces o f Mackey spaces a r e Mackey spaces. The r e s u l t i s n o t n e c e s s a r i l y t r u e i f t h e c o d i n e n s i o n of t h e subspace F i n t h e space E i s n o t f i n i t e : i n deed, s e t E:=l
00
and F:=co. C l e a r l y E i s a Mackey space and F 1 1 ) ) f (F,m(F,l ) ) , s i n c e o t h e r w i s e subspace o f E, b u t (F,m(E,l 1 1 1 u n i t b a l l B o f 1 would be compact i n ( 1 ,s(1 ,E)) and hence 1 ) ) thou@ n o t r e f l e x i v e , a c o n t r a d i c t i o n . Moreover, (F,m(E,l
i s a closed the closed 1 1 would be a !lackey spa-
ce can be c o n s i d e r e d as t h e c o u n t a b l e i n t e r s e c t i o n o f Plackey spaces: indeed, 1 1 s i n c e ( 1 , b ( l ,E)) i s separable, we have t h a t (F',s(F',E/F)) i s separable and 4.4.19
( b ) shows t h a t F i s t h e i n t e r s e c t i o n o f a d e c r e a s i n g sequence o f
c l o s e d f i n i t e - c o d i m e n s i o n a l subspaces o f E which a r e c l e a r l y Mackey soaces. I f S: denotes t h e f a m i l y o f a l l Mackey spaces i n 4.5.3
(ii),
and 4.5.2
t h e n we g e t t h e e x i s t e n c e o f a dense hyperplane o f a Mackey space which i s n o t a Mackey space. There i s a v e r y s i m p l e way o f c o n s t r u c t i n g e x p l i c i t axamples o f t h i s t y p e : l e t ( F , t ) F " \ F. Set E:= s p ( F U 4 x ) ) .
be a n o n - r e f l e x i v e F r g c h e t space and x i n
Since F i s dense i n (F",s(F",F')),
dual p a i r and F i s dense i n (E,m(E,F')). and hence ( F , t )
#
Thus (F,m(E,F'))
(E,F')
is a
i s n o t conplete
(F,m(E,F')).
On t h e o t h e r hand, i f G i s a Frgchet-Monte1 space, s e t E:=(G',m(G',G)) and l e t F be a dense subspace o f E. E i s a Mackey space and (F,m(G',G)) a l s o a Mackey space: indeed, s i n c e m(F,G)
i s f i n e r t h a n m(G',G)
is
on F i t i s
enough t o prove t h a t g i v e n any a b s o l u t e l y convex compact s e t A o f (G,s(G,F)) then A i s compact i n (G,s(G,G')).
Since A i s bounded i n (G,s(G,F))
n o n i c a l i n j e c t i o n J:GA-----(G,m(G,F)) c l o s e d in GAx(G.m(G,F))
t h e ca-
i s c o n t i n u o u s and hence i t s graph i s
and, a f o r t i o r i , i n GAx(G,m(G,G')),
which
c h e t space. A c c o r d i n g t o t h e c l a s s i c a l c l o s e d graph theorem, J:GA
is a Fr6-
-------
i s continuous and A i s bounded i n (G,s(G,G')). Since G i s -+(G,m(G,G')) Montel, A i s r e l a t i v e l y compact i n (G,s(G,G')). Since A i s c l o s e d i n (G, s(G,F)),
i t i s a l s o c l o s e d i n (G,s(G,G')).
Thus A i s compact i n (G,s(G,G'f).
Observe t h a t i f F i s a dense hyperplane o f (E',s(E',E))
f o r a Banach
space E, i t i s n o t n e c e s s a r i l y t r u e t h a t e v e r y compact s e t i n (E,s(E,F)) compact i n (E,s(E,E')):
i n RO,p.133
t r u c t e d such t h a t G ' and G" a r e separable and dim(G"/G) F:=G.
C l e a r l y F i s dense i n (E',s(E',E))
compact i n (E,s(E,F)).
is
a s e p a r a b l e Banach space G i s cons=
1. Set E:=G' and
and t h e c l o s e d u n i t b a l l B o f E i s
IfB i s compact i n (E,s(E,E'))
it follows that E i s
BARRELLED LOCALLY CONVEXSPACES
162
r e f l e x i v e and hence F i s r e f l e x i v e , a c o n t r a d i c t i o n . T h i s example a l l o w s us t o show t h a t t h e t o p o l o g y s(G',G") a b a s i s o f 0-nghbs which a r e c l o s e d i n (G',s(G',G)):
does n o t have
a c c o r d i n g t o K1,918.4.
( 4 ) b ) i t i s enough t o e x h i b i t a s e q u e n t i a l l y complete s e t i n (G',s(G',G)) which i s n o t s e q u e n t i a l l y complete i n (G',s(G',G")). (G',s(G',G))
Since 6 i s b a r r e l l e d ,
i s s e q u e n t i a l l y complete and so i s t h e c l o s e d u n i t b a l l B o f
G I . Since B i s bounded i n (G',s(G',G")),
B i s precompact i n i t . bforeover
(B,s(G' ,GI1)) i s m e t r i z a b l e s i n c e G" i s seDarable and 4.4.12 a D p l i e s . Should (B,s(G',G"))
be s e q u e n t i a l l y complete, i t would f o l l o w t h a t (B,s(G',G"))
were compact and t h i s would l e a d t o a c o n t r a d i c t i o n as above. Now we c h a r a c t e r i z e those F r e c h e t spaces E such t h a t e v e r y dense hyperp l a n e F o f (E',m(E',E)) Theorem 5.2.1:
i s a Mackey space.
L e t E be a Mackey space such t h a t ( E ' , s ( E ' , E ) )
complete. I f (E',s(E',E))
i s locally
i s n o t s e q u e n t i a l l y complete, t h e n t h e r e i s a
dense hyperplane F o f E which i s n o t a Mackey space. P r o o f : L e t ( v ( n):n=1,2,. s(E',E)) s(G,E))
. ) be a non-convergent Cauchy sequence i n ( E l ,
and v i t s l i m i t i n (E*,s(E*,E)).
We s e t G : = s p ( E ' u { v \ ) .
Then ( G ,
i s l o c a l l y complete because i t c o n t a i n s t h e l o c a l l y complete hyper-
p l a n e E (5.1.13).
Since (v(n)-v:n=1,2,..)
we a p p l y 5.1.11 t o g e t t h a t t h e
i s a n u l l sequence i n (G,s(G,E)),
c l o s e d a b s o l u t e l y convex h u l l B o f ( v ( n ) :
n=1,2,. . ) i s compact i n (G,s(G,E)). 1 We s h a l l see t h a t F:=v- ( 0 ) is t h e d e s i r e d hyperplane o f E. F i r s t we prove t h a t B n E ' i s compact i n (E',s(E',F)).To a r l y s(E',F)-precompact,
i t i s enough t o show t h a t i t i s c o n p l e t e . L e t
( w ( j ) : j c J ) be a Cauchy n e t i n ( B n E ' , s ( E ' , F ) ) s(F*,F).
Since B i s compact i n (G,s(G,E))
t h e n e t ( w ( j ) : j c J ) . We can w r i t e w ' XE
do t h i s , s i n c e B A E ' i s c l e -
=
and W E E *
i t s l i m i t i n (F*, there i s a cluster point W ' G B o f
av+u, w i t h a c K and U E E ' .
F we have t h a t
For e v e r y
+
t h e l i m i t of t h e n e t ( w ( j ) : j c J ) i n ( E ' , s ( E ' , F ) ) . Now we prove t h a t B"nF i s a 0-nghb i n (F,m(F,E')) which i s n o t a 0-nghb i n F. T h i s w i l l ensure t h a t F i s n o t a Mackey space. Since B n E ' i s dense i n (B,s(G,E)), i t i s a l s o dense i n (B,s(G,F)), hence B"nF, which c o i n c i d e s I f BonF i s a w i t h t h e p o l a r s e t o f B n E ' i n F, i s a 0-nghb i n (F,m(F,E')). 0-nghb i n F f o r t h e induced t o p o l o g y , t h e n B", which c o n t a i n s t h e c l o s u r e i n E o f B'nF,
i s a 0-nghb i n E. T h i s i m p l i e s t h a t t h e sequence ( v ( n ) : n = l , . )
i s equicontinuous, and hence v c E ' , a c o n t r a d i c t i o n ,
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Observe t h a t taken, f . i . , E:= ( l ~ ' , m ( l W 1 , l ~ the ) ) , method of proof of 5. 2 . 1 provides exanples of t h e s i t u a t i o n considered i n 4 . 5 . 2 ( i i ) . From now on, i f A i s a subset of a space E , we s h a l l denote by A* i t s c l o s u r e in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . To obtain a converse of 5.2.1 i n t h e case of Mackey duals of Frechet spaces we need some Lemmata. Lemna 5.2.2: Let A be a subset of a space E. Let L be a dense subspace o f ( E ' , s ( E ' , E ) ) w i t h dim(E'/L) = n , such t h a t A i s compact i n (E,s(E,L)). Then dim(sp(EuA*)/E) f n. Proof: Let ( u ( 1 ) ,...,u ( n ) ) be a co-basis o f L i n E l . Let z ( p ) , p = l , . . , n , be elements of ( E l ) * such t h a t z ( p ) vanishes on L and t z ( p ) , u ( q ) > equals 1 i f p=q and 0 i f p f q . We a r e done i f we prove t h a t A* i s included i n the span of E U ( z ( l ) , . . , z ( p ) \ . Let z be any point of A* and ( x ( j ) : j E J ) a net i n A converging t o z in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since A i s compact i n (E,s(E, L ) ) , t h e r e i s a point x i n E such t h a t ( x ( j ) : j c J ) converges t o x i n (E,s(E, L ) ) . We s e t y:=z-x- z < z - x , u ( p ) > x ( p ) . I f U C L , then
Lemma 5.2.3: Let A be a d i s c i n a Frechet space E , such t h a t t h e d i k e n sion of s p ( E U A * ) / E i s equal t o 1. I f x belongs t o A*, t h e n t h e r e i s a sequence ( x ( n ) : n = 1 , 2 , . . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Proof: Since A* i s not included i n E , A i s not a weakly r e l a t i v e l y compact subset of E . By K1,§24.2.( 1) t h e r e i s a countable subset B of A such t h a t B * i s not included in E . Let y be a point i n B* not belonging t o E . Clearly sp(EuA*) = s p ( E U B * ) = s p ( E O{y\). Take any point x i n A*, then x= ay+z, w i t h a € K and z c E . Let F be t h e c l o s u r e i n E of sp(B U { z \ ) and G : = sp(FUCy1). Since ( F ' , s ( F ' , F ) ) i s separable, we can apply 2.5.18 t o obt a i n t h a t ( F ' ,s( F ' , G ) ) i s separable, hence ( B * , s ( G , F ' ) ) i s m t r i z a b l e . Thus t h e r e i s a sequence ( y ( n ) : n = 1 , 2 , . . ) i n B converging t o y i n (G,s(G,F')). Now s e t t i n g x ( n ) : = ay(n) + z , n = 1 , 2 , ..., we obtain t h e desired sequence i n E converging t o x i n ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . , / Theorem 5.2.4: Let E be a Frechet space. Every hyperplane of (E',m(E',E)) i s a Mackey space i f and only i f ( E , s ( E , E ' ) ) is s e q u e n t i a l l y complete.
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Proof: I f ( E , s ( E , E ' ) ) i s not s e q u e n t i a l l y complete, we apply 5.2.1 t o obtain a hyperplane of E' which i s not a Mackey space. Conversely, suppose t h a t ( E , s ( E , E ' ) ) i s s e q u e n t i a l l y complete. Let F be a hyperplane of E ' . If F i s closed i n ( E ' , m ( E ' , E ) ) , then i t i s c l e a r l y a Mackey space. If F i s dense, i t i s enough t o show t h a t every compact d i s c in ( E , s ( E , F ) ) i s a l s o compact i n ( E , s ( E , E ' ) ) . To prove t h i s , l e t 63 be the c l a s s of a l l compact d i s c s in ( E , s ( E , F ) ) , and l e t t be t h e l o c a l l y convex topology on E such t h a t ( E , t ) i s t h e inductive l i m i t of t h e family of Banach spaces (Eg:BEB ) . Since t i s f i n e r than m ( E , F ) , t h e i d e n t i t y mapping I:(E,t)----(E,m(E,F)) i s continuous. On t h e other hand, m ( E , F ) i s coarser than t h e i n i t i a l topology of t h e FrGchet space E , hence I : ( E , t ) - - - + E has closed graph i n ( E , t ) * E . By 1.2.19 we obtain t h a t I is continuous, t h e r e f o r e B i s a bounded subset of E
f o r every B C 8 . We apply 5.2.2 t o obtain t h a t d i m ( s p ( E U B * ) / E ) 5 1 f o r every BE^ . According t o 5.2.3, i f xcB* t h e r e i s a sequence ( x ( n ) : n = l , . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since ( E , s ( E , E ' ) ) i s sequent i a l l y complete, x belongs t o E . Therefore B* i s included i n E,and t h u s every B E 63 i s compact in (E,s( E Y E ' ) ) .
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5.3 Notes and renarks. Local convergence was considered by M A C K E Y , ( I ) and GROTHENDIECK, see G , Ch.3,4, exerc. 5 where some p a r t s of 5.1.11 appear. A space i s p-complete (polar-semireflexive o r topologically complete) i f every precompact subset of E i s r e l a t i v e l y compact ( s e e K I , p.311 and DAY, (21, p.53 r e s p e c t i v e l y ) . A space E i s r - c o n p l e t e i f every sequence s a t i s f y i n g the Cauchy condition (2.2.6) i s s u m b l e . One has t h e following chain of i m l i c a t i o n s complete+quasi-conpleteep-conplete plete.
+ sq-cormlete 3 t - c o m l e t e
=+local corr-
( 1 1 , s ( l 1 , 1 ~ ) )i s a non-p-conplete s e q u e n t i a l l y corrplete space. If E i s the r e f l e x i v e Banach space constructed by JAMES,( 1) w i t h E ' , E l ' , . . . . separab l e , i t can be shown t h a t ( E ' , s ( E ' , E " ) ) i s a non-sequentiallv complete 2conplete space ( s e e DIEROLF,( 2 ) ,p.25). The sDace ( co.s( c Q , l 1 ) ) i s a non- 'Lcomplete l o c a l l y conplete space ( j u s t consider t h e canonical u n i t v e c t o r s ! ) . Weakly 2 - c o m p l e t e spaces a r e 2 - c o n p l e t e and the weakly 2 - c o n p l e t e Banach spaces a r e p r e c i s e l y those which do n o t contain co ( s e e BESSAGA,PELCZYNSKI,(4)). Moreover, i f E i s a l o c a l l y complete space such t h a t every continuous l i n e a r operator T:C( K ) - ( E , s ( E , E ' ) ) i s compact, f o r every compact space K , then E i s weakly L-conplete ( t h a t i s , ( E , s ( E , E ' ) ) i s Z - c o m l e t e ) , see THOWAS,( 1). The c h a r a c t e r i z a t i o n of local completeness a s presented in 5.1.11 i s d u e t o DIEROLF,(5). Let us observe t h a t t h i s result f a i l s t o be true in general in t h e non-locally convex s e t t i n g ( s e e DIEROLF,(Z): s e t p : = 1 / 2 and consider
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the F-space ( l P , q ) . S e t u(n):=e(n)/f?? , e ( n ) being t h e n - t h canonical u n i t vector. ( u ( n ) : n = 1 , 2 , . . ) is a n u l l sequence i n ( P , q ) and v ( n ) : = ( n - I ) . T u ( i ) belong t o acx(u(n):n=1,2,..). Since q(v(n))’nl/I, one has t h a t acx(u(n):n= l Y Z y .. ) i s not even bounded). 5.1.15 and 5.1.16 a r e taken from MAROUINA,PEREZ CARRERAS ,( 4 ) . 5.1.26 can be seen i n DIEROLFy(5). The M.c.c. and s.M.c. were introduced i n GROTHENOIECK ,(2) where 5.1.31 i s a l s o remarked. 5.1.32 i s due t o WEBB,(4). In the context of MAHWALD-type closed graph t h e o r e m one has t h e f o l l o w i n g remrkable result 5.3.1: (VALDIVIA,(43)) The weak dual of a space E i s l o c a l l y complete i f and only i f every l i n e a r mapDing f : E - + 1 2 with closed graph i s weakly c o n t i nuous. All t h e r e s u l t s which appear in 5.2 a r e taken from VALDIVIA, (13) where many o t h e r r e s u l t s on t h e s t a b i l i t y of Mackey spaces by subspaces a r e i n cluded. W e l i s t a few of them. 5.3.2: Every dense subspace of a r e f l e x i v e (LF)-space i s a Mackey soace. 5.3.3: Every dense subspace of a q u a s i b a r r e l l e d (DF)-space is a Mackey soace. 5.3.4: Every subspace of a countable product of nuclear (DFf-sDaces is a R i G y space.