Notations

5 A

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0

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NOTATIONS

W e always c o n s i d e r v e c t o r spaces over t h e complex

C, b u t a l l o u r r e s u l t s , w i t h obvious a d a p t a t i o n s , a r e v a l i d f o r real vector spaces. For i n t e r v a l s w e use t h e u s u a l n o t a t i o n , ]a,.[ , ( c , d ] e t c . l c , d l , where c < d , d e n o t e s any of t h e i n t e r v a l s ] c , d [ , ] c , d ) , field

[ c , d [ and [ c , d ) ; ( c , d ) denotes t h e i n t e r v a l ( c , d ) i f c s d and t h e i n t e r v a l [ d , c ) i f d s c. Given r e a l numbers s , t w e w r i t e s A t = i n d ( s , t ) and

--

s v t = 6LLP(S,t). h ( t , s ) E Y , f o r every Given a f u n c t i o n h: ( t , s ) E B x A t E B, ht d e n o t e s t h e f u n c t i o n S E A h ( t , s )E Y and f o r every S E A , hs denotes t h e f u n c t i o n t E B - h ( t , s ) E Y. Given a f u n c t i o n f : X v Y and A C X I f denotes t h e IA r e s t r i c t i o n of f t o A . Ix d e n o t e s t h e i d e n t i c a l automorphism of X . Given an A c X , xA denotes t h e c h a r a c t e r i s t i c f u n c t i o n of A: x ( x ) = 1 i f x E A and x A ( x ) = 0 i f x E X A and x @ A . Y : R -IR denotes t h e Heaviside f u n c t i o n Y =

x [ ~ , W~ e (d .e f i n e

sg: R

-

{-l,O,l}

by

sg t = 1 i f

= 0 f o r t = 0 and = -1 i f t < o . I f X and Y are t o p o l o g i c a l s p a c e s , E ( X , Y ) denotes t h e s e t of a l l c o n t i n u o u s f u n c t i o n s of X i n t o Y. I f a sequence xn converges t o x i n a t o p o l o g i c a l space x, w e t > O ,

write

x - xX n

x - x . n I f t h e sequence t n E R t e n d s t o t and i s d e c r e a s i n g w e w r i t e t G t ; i n an analogous way w e d e f i n e t n + t and 6 G O . For c~:~)a,b) X I a ( t - ) denotes t h e l i m i t a t t h e l e f t , when i t e x i s t s . I n an analogous way w e d e f i n e c r ( t + ) . where X i s a normed s p a c e , I l f ( 111 Given f : [ a , b ] - X I d e n o t e s t h e f u n c t i o n t E (alb] Ilf (t)II€ IR+ and u n l e s s (t)ll 1 a ,< t < b). o t h e r w i s e s p e c i f i e d If 11 denotes Aup The n o t i o n of summable series i s d e f i n e d i n t h e s e n s e of Bourbaki

-

.

or

-

{]If

NOTAT I O N S

2

-

B

Given a c l o s e d i n t e r v a l ( a , b ) c R

Id( = n

and

> 0 w e w r i t e DE = { d E D i s a f i l t e r b a s i s on D. E

Given two d i v i s i o n s d i v i s i o n of

[a,b)

dlld2E ID

write

I

Ad = AUp{lti-ti-ll

i = 1,2

I

Ad <

€1;

,..., Id[>. D(a,bb' U,={IDE

the class

o b t a i n e d by a l l p o i n t s of

w e say t h a t

d l < d 2 , i f every p o i n t of

E

and d2.

i s an o r d e r r e l a t i o n on D t h a t makes it t e r e d on t h e r i g h t . For every d € D w e d e f i n e

relation

Dd = { d ' E D

vG

the class

= {Dd

I

d€D)

f i n e r than t h e f i l t e r b a s i s 0.1

-

I

the

we The fil-

d,
i s a f i l t e r b a s i s on ID which i s V , and hence

L e t x b e a topoLogicaL bpace and f : D -X; the exibtence 0 6 L i m f ( d ) , t h a t i b t h e Limit o u c h t h e @-

t e h babia

Ad+O

VA, impLieb t h e exibtence 0 6

i b , t h e limit oveh t h e { i L t e h equal. If ACX

> 0)

d2.

dl, and

i s a p o i n t of

dl

<

I

or

d l v d2

dl

i s {Lneh t h a n

d2

(a,b]

w e denote by

dlld2EID

(a,b)

We write

D , denotes t h e s e t of a l l d i v i s i o n s of

o r simply

Given

.. . < t n=b.

d: t 0=a < t l
i s a f i n i t e sequence

a d i v i b i o n of

X

we d e f i n e t h e o b c i l a t i o n of

or

[ a , b) i = 1,2,.

U and

l a5 4=

..,Id1 . 0'

f:

f

WA(f) = b u p { l l f ( t ) - f ( S ) I I

V

VG , and b o t h

babin

i s a seminormed space and

on

I

Lim f ( d ) , t h a t

dsID

(a,b]-

,...,5 I d / 1

with

X I for

A

s,tEA);

denotes t h e s e t of a l l p a i r s

(5,

ahe

( d , O where

CiE [ti-l,ti],

denotes t h z s e t of a l l p a i r s

(d,E')

NOTATIONS where

dED

5'

and

=

(5;

-

,...,5 'Id1 )

with

3 <~E)ti-llti(.

W e s a y t h a t a f u n c t i o n f: (a,b) X is a step f u n c t i o n , w e w r i t e f E E ( [ a , b ] , X ) , i f t h e r e exists a d e n s u c h t h a t f i s c o n s t a n t i n ) t t - l I t i [ , i = 1,2,. ,I d / .

-

C

A b i e i n e a h thipk?e (BT) i s a s e t of t h r e e v e c t o r spaces F

mapping

+G; w e w r i t e

by

ExF

B:

(E,F,G)B

BT (E,F,G) where

and

are Banach s p a c e s , w i t h a b i l i n e a r x - y = B ( x , y ) and d e n o t e t h e

E l F, G , where BT

..

G

(E,F,G). A X o p o e o g i c d BT i s

o r simply E

t o o i s a normed s p a c e and

nuous ; w e suppose t h a t

B

a

is conti-

11 B I I < 1.

E X A M P L E S - L e t W, X and Y b e Banach s p a c e s . 1. E = L ( X , Y ) , F = X I G = Y and B ( u , x ) = u ( x ) . 2. E = L ( X , Y ) , F = L(W,X), G = L ( W , Y ) and B ( v , u ) = v o u .

0.2

-

3.

E = Y, F = Y',

4.

E = G = Y,

G = C

F = C

and

and

B(y,yl) =(y,y*).

B(y,X) = Xy.

a ) E x . 1 i s a p a r t i c u l a r i n s t a n c e of E x . 2 :

take

b ) Ex.3 i s a p a r t i c u l a r i n s t a n c e of E x . 2 :

t a k e X = C and

w

= Y.

c ) Ex.4 i s a p a r t i c u l a r i n s t a n c e of Ex.2: Given a BT ( E , F, G ) B IIXllB =

f o r every

EB = E x E E

EB

w i t h t h e norm

l o g i c a l BT (EB,F,G) D

-

Let

d e f i n e d on

E E

I

xEE

I

AuPE IIB(X,Y)ll

and

w e endow

t a k e X = W = C.

we define

IIYII .c< 1)

IIx\lB
11 [ I B

and w e s a y t h a t t h e

topo-

i s a b b o c i a t e d t o t h e BT ( E , F , G ) .

b e a v e c t o r s p a c e and such t h a t

plI...,pm~

rE

b ~ p t P l , . . . r ~ m ] ErE.

rE

W = C.

d e f i n e s a t o p o l o g y on

E: t h e sets

rE

a s e t of s e m i n o r m

implies

NOTATIONS

4

form a b a s i s of neighborhoods of 0 ; t h e s e t s xo+v form PIE a b a s i s of neighborhoods of xo€ E . Endowed w i t h t h i s topology E i s called a l o c a l l y convex space (LCS). A LCS E i s s e p a r a t e d i f and o n l y i f p ( x ) = 0 f o r every P E r implies x = 0. E A sequence xn of a LCS E i s c a l l e d a Cauchy beqUenCe i f f o r every P E TE and every E > 0 t h e r e e x i s t s an nE(p) such t h a t f o r n,m >,n,(p) w e have p(x,-x,) < E. A separated s e q u e n t i a l l y complete LCS (SSCLCS) i s a s e p a r a t e d LCS inwhich every Cauchy sequence i s convergent. A F r e c h e t space i s a SSCLCS whose topology can be d e f i n e d by a c o u n t a b l e s e t o f seminorms (and i s t h e r e f o r e m e t r i s a b l e )

.

EXAMPLES LCS 1 - Every normed o r seminormed space E i s a LCS. LCS 2 - I f X i s a LCS and K a compact space there i s a n a t u r a l s t r u c t u r e of LCS on E = & ( K , X ) : f o r every seminorm PE rx w e d e f i n e a seminorm p E r E by p ( f ) = b u p p [ f ( t ) ] , where

f E E = 6 ( K , X ) ; w e o b t a i n on

6(K,X)

tEK

t h e topology of

uniform convergence on K . I f X i s a Banach o r a F r e c h e t space, so i s 6 ( K , X ) . LCS 3 - L e t X be a normed space; E = 6 ( )a , b [ , X ) becomes a LCS when endowed w i t h t h e family of seminorms

where [c,d) runs over a l l c l o s e d i n t e r v a l s of ] a , b ( . I f X i s complete, i . e . a Banach space, E i s a F r e e h e t space; i t s topology may be d e f i n e d by t h e countable s e t of seminorms , n€m. I' "[a+:, b-

i)

LCS 4

-

Let

X

be a LCS; E = &(]a,b[,

X)

becomes

t u r a l l y a LCS when endowed with t h e family of seminorms P(c,d] where

PE

rx

and

[f]

= bup{p[f(t)]

[c,d]

C

] a , b [.

I

c < t ( dl

na-

NOTATIONS For LCS

and

X

Y,

5

d e n o t e s t h e v e c t o r s p a c e of

L(X,Y)

a l l c o n t i n u o u s l i n e a r mappings from

X i n t o Y ; i n o r d e r that a l i n e a r mapping f : X 4 Y be c o n t i n u o u s i t i s s u f f i c i e n t t h a t it i s c o n t i n u o u s a t t h e o r i g i n and hence f o r e v e r y q e r y

there is a every

PE

rX

X

i s a normed s p a c e and

a >0

and

such t h a t

q[f ( x ) ] 6 a p ( x )

for

xEX.

-

LCS 5

If

L(X,Y) we

a LCS, on

Y

c o n s i d e r t h e t o p o l o g y d e f i n e d by t h e seminorms p ( f ) = bUp{P(f

Y

If

I

(XI)

i s a SSCLCS s o i s

XEX,

L(X,Y).

-

I f X i s a LCS and f : [a,b) d e f i n e t h e o s c i l a t i o n s as i n B: f o r w

deD

and f o r w

qrA

q,i

IIxII G 1 1 ,

PE

ry. q € Tx w e

f o r every A C X we w r i t e

( f ) = bUp{q(f(t)-f(S))

X

I

tlsEAl

we w r i t e

(f) = w

(f)

(ti-1*ti)

W;lIi(f)

I

etc.

-

= w

q I ) ti-1 I t i

[ (f)

I

A t o c a e e y convex BT (LCBT) i s a s e t o f t h r e e v e c t o r E l F , G , where F i s a Banach s p a c e , G a SSCLCS,

spaces

w i t h a b i l i n e a r mapping E

-

B: ExF

G.

-

T H E O R E M 0.3

Let

X

b e a c o m p l e t e m e t h i c bpace and

p o t o g i e a l &pace. Foh ewehy

i E I

te2

Ti:

that: 1)

-

I n c h a p t e r I11 w e w i l l u s e the f o l l o w i n g

X

X

a&

I

be

huch

LocaLty a unidahm c o n t h a c t i o n , i . e . , d o h evehy ioE I t h e h e k b a neighbohhood J and a c o n b t a n t c J < 1 buch t h a t (Ti)iEI

i6

d[TiXr

doh

aeL

2 ) Foh e v e h y

x,yEX XE X

tinuoua. T h e n , id d o h evehy

TiyJ

S

cJd(xry)

and ewehy i E J . t h e d u n c t i o n i €I XEX,

xi

MT

ixEX

id

denotes t h e d i x e d p o i n t

con06

Ti

6

NOTATIONS

-

( w h i c h e x i b t d b y Banach c o n t h a c t i o n mapping t h e o h e m ] , t h e mapping i € I xi€ x i4 c o n t i n u o u b . P R O O F . Obviously i t i s enough t o prove t h a t t h e mapping i s continuous a t

i

0

. W e have

d(xi8xi

) 0

= d[Tixi

s

0

cJd (xi

0

] +

d[Tixi

s

]

i

TiXio I

x

iO

d[Tixi

+ d[TiXi

0

0

)

0

s 1-c d[Tixi

t h a t by 2 ) goes t o zero when

i+io.

0

,T.

xi ]

lo

0

0

xi ]

0

‘Ti X i

hence d(xi8xi

‘Ti

0

]

0