SOME

THEOREMS

WITH

A FUNCTIONAL V.

M.

ON S E M I C O N T I N U I T Y

AND

CONVERGENCE

Gol'dshtein

UDC 513.881

We s h a l l c o n s i d e r f u n c t i o n a l s F (which m a y take on the v a l u e ~) o v e r s o m e l o c a l l y c o n v e x m e t r i c s p a c e X. In t h i s a r t i c l e we s h a l l u s e F : X ~ (R, o0) as the s t a n d a r d n o t a t i o n f o r such f u n c t i o n a l s . The f u n d a m e n t a l r e s u l t i s a t h e o r e m which s t a t e s the following: s u p p o s e that we a r e g i v e n a s e q u e n c e offunctionals F re:X(R, ~) ( m = 1 , 2 ) s u c h that w h e n m ~ ~o, we have F m ~ F 0 u n i f o r m l y on a n y b o u n d e d s e t in X. Let {Xm} be a s e q u e n c e of p o i n t s of the s p a c e X which c o n v e r g e s w e a k l y to s o m e p o i n t x 0 E X . W e a s s u m e that f o r m ~ o0 we have Fm(x~,) -+Fo(xo) < c o .

(1)

We s h a l l e s t a b l i s h that g i v e n c e r t a i n c o n d i t i o n s , the fact that the r e l a t i o n (1) is s a t i s f i e d g u a r a n t e e s the i n e q u a l i t y lira Km(x,~) ~ Ko(xo) r n ---~- ~

w i l l be s a t i s f i e d for a n y s e q u e n c e of f u n e t i o n a l s {Km} which, when m ~ ~, c o n v e r g e s u n i f o r m l y to K 0 on a n y b o u n d e d s e t i n X and which c o n s i s t s of f u n c t i o n a l s m a j o r i z e d f r o m below in s o m e s e n s e by the f u n c t i o n a l s {Fm}. T h e following r e s t r i c t i o n s a r e p l a c e d on the f u n c t i o n a l s F m , F0: 1) the F m a r e c o n v e x ; 2) F 0 is s t r o n g l y c o n v e x in the s e n s e of the d e f i n i t i o n g i v e n i n S e c t i o n 2. In a d d i t i o n , c e r t a i n r e s t r i c t i o n s a r e p l a c e d on the o r d e r of m a g n i t u d e of the f u n c t i o n a l s F m. In the f i r s t p a r t of this a r t i c l e we s h a l l p r o v e a t h e o r e m on s e m i c o n t i n u i t y for the c o n v e x f u n c t i o n a l s Fro: X ~ (R, ~), w h e r e X is a g e n e r a l l o c a l l y c o n v e x s p a c e . T h i s will be u s e d as a n a u x i l i a r y t h e o r e m . In the s e c o n d p a r t we s h a l l p r o v e the m a i n t h e o r e m . We s h a l l c o n s i d e r s e p a r a t e l y the c a s e in which X i s a r e f l e x i v e m e t r i z a b l e s p a c e o r one which is c l o s e to b e i n g r e f l e x i v e . T h e m a i n t h e o r e m is a g e n e r a l i z a t i o n of the c o r r e s p o n d i n g t h e o r e m g i v e n i n [1] for the c a s e of i n f i n i t e - d i m e n s i o n a l s p a c e s and f u n c t i o n a l s not r e p r e s e n t a b l e by i n t e g r a l s . The idea of a p p r o x i m a t i n g a f u n c t i o n a l by c o n v e x f u n c t i o n a l s a n d t h e r e a f t e r a p p l y i n g the t h e o r e m on s e m i c o n t i n u i t y has b e e n t a k e n f r o m [11. F o r the c a s e of one f u n c t i o n the t h e o r e m s of [1] follow f r o m o u r r e s u l t s ; for a s e q u e n c e of f u n c t i o n s , t h e y will follow only if c e r t a i n r e s t r i c t i o n s a r e i m p o s e d .

I.

THEOREM

ON

SEMICONTINUITY

H e r e i n a f t e r R will denote a real straight line. Let us fix s o m e l o c a l l y c o n v e x s p a c e X. We c a l l a l o c a l l y c o n v e x s p a c e b o r n o l o g i c a l if in this s p a c e e v e r y l i n e a r f u n c t i o n a l b o u n d e d on e v e r y b o u n d e d s e t is c o n t i n u o u s . Suppose t h a t the s p a c e X is b o r n o l o g i c a l . Let M be a n a r b i t r a r y s e t i n X and l e t x 0 E M. We s h a l l c a l l the point x 0 a C - i n t e r i o r point of the s e t M if f o r e v e r y p o i n t x E X\{x0} t h e r e e x i s t s an a 0 > 0 s u c h t h a t for e v e r y a E ((~0,0] the point y = a ( x - x 0) + Xo~, M. .. . . . . . . . . . . . . , _ T r a n s l a t e d f r o m S i b i r s k i i M a t e m a t i c h e s k i i Z h u r n a l , Vol. 12, No. 1, pp. 84-98, J a n u a r y - F e b r u a r y , 1971. O r i g i n a l a r t i c l e s u b m i t t e d A p r i l 7, 1969. 9 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

60

The f u n c t i o n a l

l(x) ---=g(x) @ K,

(1.1)

w h e r e g i s a l i n e a r f u n c t i o n a l on X and K is a r e a l n u m b e r , w i l l be c a l l e d a h y p e r s u r f a c e fmactional. Let 1 be a h y p e r s u r f a c e f u n c t i o n a l . surface.

The s e t r l = { (x, y).. y = Z(x)} in X • R will be c a l l e d a h y p e r -

O b v i o u s l y the s e t of a l l h y p e r s u r f a c e f u n e t i o n a l s is a l i n e a r s p a c e . Let X be the l i n e a r s p a c e of a l l l i n e a r f u n c t i o n a l s on X. By v i r t u e of (1.1), to e v e r y h y p e r s u r f a e e f u n c t i o n a l t h e r e c o r r e s p o n d s a p o i n t in the s p a c e L = X • R a n d v i c e v e r s a . Suppose that we a r e g i v e n a s e t A E "~. will be d e n o t e d by L A.

The s e t of h y p e r s u r f a e e f u n e t i o n a l s c o r r e s p o n d i n g to A x R

Let us c o n s i d e r the f u n c t i o n a I F :X ~ (R, ~ ) ~ t h e r e e x i s t s a s u b s e t Lt c L A s u c h that

F ( x ) ~ sup l(x)

We s h a l l c a l l it c o n v e x o v e r L A at the p o i n t x 0 E X, if

for all x ~ X and F(xo) ~---sup l(x0).

ZELt

A functional F : X ~ p o i n t x o.

(1.2)

~LI

(R, ~ ) w h i c h is c o n v e x o v e r X at the p o i n t x 0 E X, w i l l be c a ! l e d c o n v e x at the

A f u n c t i o n a l F w h i c h is c o n v e x o v e r LA at e v e r y point x E X w i l l be c a l l e d e o n v e x e v e r L A. O b v i o u s l y the l a s t d e f i n i t i o n c a n be t r a n s f o r m e d a s f o l l o w s : a f u n c t i o n a l F w i l l b e c a l l e d c o n v e x o v e r L A if t h e r e e x i s t s a s u b s e t L 1 c L A s u c h that

F(x)-~-supl(x)

for all x C X .

(1.3)

IE L 1

z If A = X, we o b t a i n t h e u s u a l d e f i n i t i o n of a c o n v e x f u n c t i o n a l , We s h a l l s a y that the f u n c t i o n a l F : X ~ (R, ~) i s s e m i c o n t i n u o a s f r o m b e l o w a t a p o i n t x 0 E X if for e v e r y r e a l N < F(x0) t h e r e e x i s t s a n e i g h b o r h o o d V(x0) of the p o i n t x 0 s u c h that F(x) > N for all x E V(x0). A f u n c t i o n a l F : X ~ (R, ~) w i l l be e a l I e d s e m i c o n t i n u o u s f r o m below if it i s s e m i e o n t i n u o u s f r o m beIow at e v e r y p o i n t x E X. Let the s p a c e X ' be c o n j u g a t e to X, and l e t X 0' b e a l i n e a r s u b s p a e e of X'. The s p a c e X with the weak topology g e n e r a t e d by X0'* w i l i be w r i t t e n a s (X, Crx0'); if X 0 ' = X ' , we s h a l l w r i t e s i m p l y (X, ~). Let us c o n s i d e r the s e q u e n c e of f u n e t i o n a i s F m : X ~ (R, ~). We s h a l l c a l l it u n i f o r r n l y e o n v e x o v e r LX 0' at the p o i n t x 0 E X ( s e q u e n t i a l l y u n i f o r m l y c o n v e x o v e r L X ' at the point x 0 E X) if for a n y r e a l N < t i m 0 Fm(x0) (m ~ ~) t h e r e e x i s t s a s e q u e n c e of h y p e r s u r f a c e f u n e t i o n a l s {lm : 1m ( LX0, } w h i e h is e q u i c o n t i n u o u s at the point x 0 i n the s p a c e (X, ox0, ) and is s u c h that lira t n (Xo) ~ N, F ~ (x) ~ l,~ (x) for all x ~ X. 7n~oo

( T h e r e e x i s t s a s e q u e n c e of h y p e r s u r f a c e f u n c t i o n a l s . { l m : l m E LX0 ~} s u c h that .l.i.r.a / m ( X m ) _> N for any seqfience of p o i n t s {x m : x m E X} that c o n v e r g e s to x 0 at F m ( x ) _> /re(x) (x, aX0 ') a n d for a l l xEX.) A s e q u e n c e of f u n c t i o n a l s F m : X ~ (R, ~ ) which is u n i f o r m l y c o n v e x o v e r LX0, ( s e q u e n t i a l l y u n i f o r m ly c o n v e x o v e r L X ,) at e v e r y p o i n t x E X w i l l be c a l l e d u n i f o r m l y e o n v e x o v e r LX0' ( s e q u e n t i a l l y u n i f o r m l y c o n v e x o v e r LX0,) 0 A s e q u e n c e of f u n c t i o n a l s which is u n i f o r m l y c o n v e x o v e r L X, ( s e q u e n t i a l l y u n i f o r m l y c o n v e x o v e r LX,) will be c a l l e d w e a k l y u n i f o r m l y c o n v e x ( s e q u e n t i a l l y w e a k l y u n i f o r m l y convex). LEMMA 1. A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r the f u n c t i o n a l F : X ~ (R, ~) to be c o n v e x o v e r L X, at the p o i n t x 0 E X is that it be c o n v e x at the p o i n t x 0 and s e m i c o n t i n u o u s f r o m b e l o w at t h i s s a m e point. *We shall use the term "the weak topology generated by X0' " to denote the weaker of the topologies on X for which all functionals f @X0' are continuous.

61

P r o o f . N e c e s s i t y . Since the f u n c t i o n a l f is c o n v e x o v e r L X, at the point x 0 , it is o b v i o u s l y c o n v e x at the point x 0 . We s h a l l now p r o v e that the f u n c t i o n a l F i s s e m i e o n t i n u o u s f r o m below at the p o i n t x 0 . Since X i s c o n v e x o v e r L X ' at the p o i n t x 0 , it follows that t h e r e e x i s t s a s e t L 1 c L X, s u c h that

F(x) ~

sup t(x)

(1.4)

IEL1

for all x E X and

F(Xo) =

sup l(z0).

(1.5)

IEL I

Let us a r b i t r a r i l y s e l e c t N < F(x 0) and ~ > 0. Let l E L 1 be a f u n c t i o n a l such that/(x0) > N. s e l e c t a n e i g h b o r h o o d V(x0) of the p o i n t x 0 i n which we will have

IZ(x) - Z(x0) I < ~ for allxEV(X0)o

F r o m (1.4) and (1.6) we have F(x) > /(x) > / ( x 0 ) - r

We

(1.6) > N-r

Since N a n d r w e r e c h o s e n a r b i t r a r i l y , this p r o v e s the s e m i c o n t i n u i t y of the f u n c t i o n a l F. S u f f i c i e n c y . A s s u m e t h a t the f u n c t i o n a l F is c o n v e x at the p o i n t x 0 and is s e m i c o n t i n u o u s f r o m b e l o w at t h i s point. Since the s p a c e X is l o c a l l y c o n v e x , it follows that t h e r e e x i s t s i n X a f u n d a m e n t a l s y s t e m of c o n v e x n e i g h b o r h o o d s . Let us take a n a r b i t r a r y r e a l n u m b e r N < F(x0). Since the f u n c t i o n a l F is s e m i c o n t i n u o u s f r o m below at the p o i n t x 0, we c a n s e l e c t a c o n v e x n e i g h b o r h o o d V(x0) of this p o i n t s u c h t h a t F(x) > N for all x E V(x0). Let us c o n s i d e r the s p a c e X • R.

Let A be a n a r b i t r a r y s e t i n X x R.

We s h a l l d e n o t e its c o n v e x

hull by vA. In X x R we c o n s t r u c t the s e t V={(x,g):xEV(xo),y:N--i},Ao=VU(x0,

N),

(1.7)

AF = {(z; g) : y > F(x)}. Let v A. be the c o n v e x h u l l of the s e t A 0. We s h a l l p r o v e that the s e t vA0 h a s a C - i n t e r i o r point. Let us c o n s i d e r t~e p o i n t (x 0, N - 1 / 2 ) . We s h a l l p r o v e that it i s a C - i n t e r i o r p o i n t of the s e t VA0. We note that the l a s t s t a t e m e n t is e q u i v a l e n t to the following: let VA^ be an a r b i t r a r y t w o - d i m e n s i o n a l s e c t i o n of the s e t VA0 b y the p l a n e L which is p a r a l l e l to the line x = 0 a n ~ p a s s e s t h r o u g h the p o i n t (x 0, N - 1/2). T h e n the p o i n t (x 0, N - - 1 / 2 ) i s an i n t e r i o r p o i n t of the s e c t i o n v A. Since the s e t s AF,VA0 a r e c o n v e x and s i n c e VA0 has a C - i n t e r i o r point, we have s a t i s f i e d the c o n d i t i o n s of the t h e o r e m on s e p a r a b i l i t y ([2], p. 446), f r o m which it follows that t h e r e e x i s t s a h y p e r s u r f a c e F l which s e p a r a t e s A F and VA0. Let l be the h y p e r s u r f a c e f u n c t i o n a l c o r r e s p o n d i n g to the h y p e r s u r f a c e F l. We s h a l l p r o v e that l E LX,. By the d e f i n i t i o n of h y p e r s u r f a c e f u n c t i o n a l , we have /(x) = g(x) + K, w h e r e the f u n c t i o n a l g is l i n e a r andKER. It follows f r o m (1.5) t h a t / ( x ) = g ( x ) + K >NforallxEV(x0). M o r e o v e r , g(x) > N - - K f o r a l l x E V(x0). Let g(x) = g ( x - x 0 ) ; t h e n ~(x) > N - K for a l l x E V(x0). F r o m t h i s , s i n c e g is l i n e a r , it follows t h a t it is b o u n d e d on e v e r y b o u n d e d s e t in X. Since the s p a c e X is b o r n o l o g i c a l , the b o u n d e d n e s s of the f u n c t i o n g on e v e r y b o u n d e d s e t m e a n s that it is c o n t i n u o u s . T h e r e f o r e the f u n c t i o n a l s g and l a r e a l s o c o n t i n u o u s . F r o m t h i s , s i n c e N was c h o s e n a r b i t r a r i l y , it follows t h a t the f u n c t i o n a l F i s c o n v e x o v e r LX' at the p o i n t x 0. To show t h i s , let us c o n s i d e r the s e q u e n c e of r e a l n u m b e r s N n ( n = 1, 2 , . . . ) . Let Nn < F(x0) f o r all v a l u e s of n. A c c o r d i n g to the f o r e g o i n g d i s c u s s i o n , it is p o s s i b l e to c o n s t r u c t a s e q u e n c e of f u n c t i o n a l s l n E L X ' which, by v i r t u e of (1.7), will s a t i s f y the c o n d i t i o n s

l~ (z) ~< F (z) f o r all x E X a n d a l l n; f u r t h e r m o r e ln(X0) -> N for all n.

F(x) ~ h o l d s f o r a l l x E X.

62

F r o m (1.8) it follows that the i n e q u a l i t y

sup L(z} n=i,

(1.s)

2 ....

P a s s i n g to the l i m i t i n (1.9), we o b t a i n

(1.9)

F(xo) ~-- lim l,~(xo) ~ n-->~

for alI n.

sup l~(x,) n=i,

(1.10)

2,...

F r o m this r e l a t i o n and (1.9) it follows that

sup l.(Xo)

F(x0)=

n=i,

g...

COROLLARY. A n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n for c o n v e x i t y of the f u n c t i o n a l F : X ~ (R, ~) o v e r L X, i s that the f u n c t i o n a l b e c o n v e x and s e m i e o n t i n u o u s f r o m below. L e t us s p e c i f y s o m e l i n e a r s u b s p a e e X 0' E X'. T H E O R E M 1. Suppose t h a t the s e q u e n c e of f u n c t i o n a I s F m : X ~ (R, oo) (m = 1, 2 . . . . ) is u n i f o r m l y c o n v e x o v e r (X, aXe ') at the p o i n t x 0. T h e n for a n y s e q u e n c e x m E X (m = 1, 2 , . . . ) which c o n v e r g e s i n (X, crX0 ,) to x 0 the ~ollowing i n e q u a l i t y h o l d s : lira F ~ (x~) ~ lira F m (xo). 7n~oo

P r o o f . Let us a r b i t r a r i I y s e l e c t N < l i m Fm(X0) a s m ~ co. Fm (xo) > 2V"q- (~/~) ( lira F m (Xo)

L e t m 0 be s u c h that w h e n m > m 0 -

-

N),

(1.11)

rn~oo

Since the s e q u e n c e F m : X ~ (R, 00)is u n i f o r m l y c o n v e x at the p o i n t x 0 o v e r LX 0 ,, it follows that t h e r e e x i s t s a s e q u e n c e of f u n e t i o u a l s { / m : l m E LX0,} w h i c h is e q u i e o n t i n u o u s at the p o i n t x 0 i n the s p a c e (X, crX00 a n d is s u c h that

F~ (z) > Z~(x)

(L 12)

f o r alI x E X and f o r a l l m a n d F• (xo) - - Im (Xo)> (1/2) ( lira Fm (xo) - - N). m~c~

C o m p a r i n g the l a s t i n e q u a l i t y with (1.11), we o b t a i n lira Im (xo) > / N .

(1. !3)

Since the f u n c t i o n a l s l m a r e e q u i c o n t i n u o u s at the p o i n t x 0 i n the s p a c e (x, crX0 ,), it follows f r o m (1.12) a n d (1.13) t h a t l i m F m ( x ~ ) ~ lira l ~ ( x m ) > / N . 7rt~oo

~-~oo

Since N < F(x0) is a r b i t r a r y , t h i s p r o v e s the t h e o r e m . If Fm(X) = F(x) f o r a l l x E X a n d f o r a l l m, it follows t h a t the c o n d i t i o n of u n i f o r m c o n v e x i t y at the p o i n t x 0 o v e r LX0, r e d u c e s f o r s u c h a s e q u e n c e to the c o n d i t i o n that the f u n c t i o n a l F be c o n v e x o v e r LX0, at the p o i n t x 0. COROLLARY. If a f u n c t i o n a l is c o n v e x o v e r LX0, at the p o i n t x0, it is s e m i c o n t i n u o u s f r o m b e i o w in (X, aX0,) at the p o i n t x 0. Let X 0' = X'.

T h e n f r o m L e m m a 1 and T h e o r e m 1 we o b t a i n the f o l l o w i n g t h e o r e m .

T H E O R E M 2. If a f u n c t i o n a l is c o n v e x at the poir, t x 0 a n d s e m i c o n t i n u o u s f r o m b e l o w at that point, it i s w e a k l y s e m i c o n t i n u o u s f r o m below. COROLLARY. Any c o n v e x f u n c t i o n a l which i s s e m i c o n t i n u o u s f r o m b e l o w is w e a k l y s e m i c o n t i n u o u s f r o m below. 2.

THEOREM

ON

CONVERGENCE

1. Let us s p e c i f y s o m e t o p o l o g i c a l s p a c e R.

WITH

A FUNCTIONAL

Let C (It) be a s e t of c o n t i n u o u s r e a l f u n c t i o n s on R.

Let ~f be s o m e s u b s p a c e of C (R) w h i c h s a t i s f i e s the f o l l o w i n g c o n d i t i o n s : if f E 2 a n d k 6 R, t h e n kfE~;iffl, fzE~f, t h e n f l + f z ~ o C f ; i f M E R a n d f E ~ a ~ , t h e n M + f E _ q ? . We s h a l l s a y t h a t the f u n c t i o n g : R (R, oo) is c o n v e x o v e r ~f at the p o i n t x 0 E R if t h e r e e x i s t s a s u b s p a c e ~f 1 c ~ s u c h that

63

g (x)/> sup ] (x)

(2.1)

/EL~

for all x E R, w h e r e g(x 0) = sup f(x0). /EL~

If a f u n c t i o n is c o n v e x o v e r s at e v e r y p o i n t x E R, we s h a l l s a y t h a t it i s c o n v e x o v e r &v. T h i s d e f i n i t i o n m a y be t r a n s f o r m e d a s f o l l o w s : a f u n c t i o n g : R ~ (R, ~ ) i s c a l l e d c o n v e x o v e r ~ if t h e r e e x i s t s a s e t ~fl c ~ s u c h t h a t g(x) -- sup f(x). ]~LI

O b v i o u s l y , if a f u n c t i o n is c o n v e x o v e r .Cf at a p o i n t x 0 E R, it i s s e m i c o n t i n u o u s f r o m below at this point. We s h a l l s a y that a f u n c t i o n g : R ~ (R, ~o) is s t r o n g l y c o n v e x o v e r ~f at the point x 0 E R if t h e r e e x i s t s a f u n c t i o n f E ~ s u c h t h a t f(x0) = g(x0) a n d the i n e q u a l i t y inf

(g(x)--/(x))>O

:r R',V(~o)

h o l d s f o r e v e r y n e i g h b o r h o o d V(x0) of the p o i n t x 0. Suppose that we have a f u n c t i o n F : R ~ (R, ~); t h e n its e x a c t m i n o r a n t c o n v e x o v e r ~f at the p o i n t x 0 is d e f i n e d as F(x) = sup g(x), w h e r e the s u p r e m u m is t a k e n o v e r the s e t of a l l f u n c t i o n s g which a r e c o n v e x gg

o v e r ~f at the p o i n t x 0 and a r e such that g(x) _< F(x) for all x E R. The following a r e o b v i o u s p r o p e r t i e s of the e x a c t m i n o r a n t c o n v e x o v e r ~f at the p o i n t x0: l~(x) ___ F(x); if g(x) i s c o n v e x o v e r Sr at the point x 0 a n d g(x) __
s u c h that g(x) _< F(x) for a l l x E Ro The p r o p e r t i e s of the e x a c t m i n o r a n t c o n v e x o v e r 5r a r e a n a l o g o u s to the above p r o p e r t i e s of the e x a c t m i n o r a n t c o n v e x o v e r g~ at the point x 0 . LEMMA 2. Let the f u n c t i o n F : R ~ (R, ~r be s t r o n g l y c o n v e x o v e r ~f at s o m e point x 0 E R , l e t the f u n c t i o n G : R ~ (R, r162be s e m i c o n t i n u o u s f r o m below at t h i s point, and l e t F(x 0) < 0% G(x0) < oo We a s s u m e t h a t t h e r e e x i s t s a f u n c t i o n go E ~f w h i c h s a t i s f i e s the i n e q u a l i t y kF(x) + go (x) -< G(x) (k E R) for all x E R. T h e n for a n y )t > - k t h e r e c o r r e s p o n d s to the f u n c t i o n kF(x) + G(x) a f u n c t i o n F~(x) which is i t s e x a c t m i n o r a n t c o n v e x o v e r ~f at the p o i n t x0, a n d the f u n c t i o n Hk (x) = XF(x) + G (x) - F ~ (x) s a t i s f i e s the r e l a t i o n Hk(x) ~ 0 as X ~ ~o. P r o o f . The fact that the f u n c t i o n ~F(x) + G(x), w h e r e 7, > - k , has a n e x a c t m i n o r a n t c o n v e x o v e r ~f at the p o i n t x 0 follows i m m e d i a t e l y f r o m the i n e q u a l i t y

EF(x) -~ G(x) ~ (~. -f- k)F(x) + go(x),

(2.2)

w h i c h follows d i r e c t l y f r o m the c o n d i t i o n s of the 1 e m m a . Let us now p r o v e that Hx(x 0) ~ 0 as ~ ~ ~. We s e l e c t c > 0 a r b i t r a r i l y . Since G(x) and F(x) a r e s e m i c o n t i n u o u s f r o m below at the point x0, t h e r e e x i s t s a n e i g h b o r h o o d V(x 0) of t h i s p o i n t s u c h that Ig0(x)--g0(x0)I < a / 3 ,

G(x) > G(xo)--e/3,

F(x) > F ( x o ) - - e / 3 k

(2.3)

for all x E V(x0). Since the f u n c t i o n F i s s t r o n g l y c o n v e x o v e r s at the p o i n t x 0, t h e r e e x i s t s a f u n c t i o n f E ~ s a t i s f y i n g the r e l a t i o n F(x 0) = f(x 0) a n d the i n e q u a l i t y F(x) _> f(x), w h i c h is s a t i s f i e d for a l l x E R. F u r t h e r m o r e , for a n y n e i g h b o r h o o d V(x 0) of the p o i n t x 0, we have inf

(F(x)--/(x))>O.

(2.4)

~E a\v'(:,O

Let us a g a i n c o n s i d e r the i n e q u a l i t y (2~

a) ~F(zo) + G(Zo)

O b v i o u s l y t h e r e a r e two p o s s i b l e c a s e s : -

-

(~ § k)F(xo) - go(x0) ~< ~,

b) ~F(x0) + V(x0) - - ( ~ + k)F(x01--go(xo) > ~.

64

(2.5)

Rewriting

the inequalities and canceling

similar terms,

G ( x o ) - - k F ( x o ) - - g0(x0) : Let us consider follows that

case a).

From

e,

we obtain

G(xo) - - kF(xo) --gQ(xo) > e.

the definition of the exact minorant

convex

over ~f at the point x 0 it

~F(zo) + C(xo) > F~(xo).

(2.6)

Comparing (2.2) and (2.6) with the relation Hs(xo) = },F(xo) -t- G(xo) - - Fs(xo),

(2.7)

w e o b t a i n at t h e p o i n t x 0 t h e i n e q u a l i t y H , ( x , ) < ~F(xo) ~r- G(xo) - - ()~ -4- k)F(xo) - - go(xo) < e. Now l e t us c o n s i d e r c a s e b). '~l-~ inf We s e t d = rain (/, m ) / 2 .

(2.8)

We i n t r o d u c e t h e n o t a t i o n (F(x)--l(x)),

m-----G(xo)--kF(xo)--go(xo)--e.

Obviously the inequality F ( x ) ~ ](x) + 2d

(2.9)

h o l d s f o r a l l x E R\V(x0). We s e l e c t k 1 to be s u c h t h a t (~1 + k)d _ - k F ( x 0 ) + G(x0) - g0(x0) - ~. By v i r t u e of t h i s c h o i c e , we h a y e d k 1 > -k. L e t us c o n s i d e r t h e f u n c t i o n f ( x ) = (k l + k ) f ( x ) + (X 1 + k ) d + g 0 ( x ) . tt f o l l o w s f r o m (2.9) t h a t t h e i n e q u a l i t y

(~, + k)F(x) + go(x) > (~,~+ k)l(x) + (~ + k)d + goCx) > 7(~) h o l d s f o r a l l x E R\V(x0). (2.10) i t f o l l o w s t h a t

We s h a l l show t h a t XIF(x) + G(x) > f(x) f o r a l l x e R.

(2.10)

L e t x ~ R\V(x0); t h e n f r o m

~.~F(x) +C(~) > ( ~ + k ) F ( x ) +go(~) > ](x). If xE V(x0), t h e n f r o m (2.3) w e o b t a i n ~ F (x) -4- G (x) > ( ~ + k ) F (x) - - kF (x) + G (xo) - - e / 3 > (~,~ + k ) F (x) - - kF (xo) + G (xo) - - 2 e / 3 (~,~ + k ) / ( x ) - - k F ( x , ) + G(xo) -~- go(x) - - go(xo) - - a = ()~ + k ) ] ( x ) + ( ~ -~- k ) d -~ go(x) = f(x)o S i n c e ~ (x) i s c o n v e x o v e r ~ at t h e p o i n t x0, it f o l l o w s f r o m t h e p r o p e r t i e s of t h e e x a c t m i n o r a n t con-. v e x o v e r ~ at t h e p o i n t x 0 a n d f r o m t h e p r e c e d i n g i n e q u a l i t y t h a t Fx,(x) ~ ~(x).

(2.i1)

F r o m (2.11) and t h e r e l a t i o n

~F(Xo) + e(Xo) - f(x0) = ~,F(x0) + ~(~0) -- (~ + k)/(zo) -- (~.~+ k)d + go(x~) = Z,F(xo) + V(xo) - - (~, +

k)F(zo) - -

V ( z o ) + e - go(zo) + g o ( ~ ) =

i t f o l l o w s t h a t H),l(x0) = A1F(x0) + G(x0) - F l l ( x 0 ) < a . follows that

If H l ( x 0 ) i s a n o n i n e r e a s i n g f u n c t i o n of l . t h e n it

H,. (xo) < e for allX <~. inequality

(2,12)

To show t h i s , w e n o t e t h a t b y d e f i n i t i o n of t h e e x a c t m i n o r a n t c o n v e x ~ a t

XF(x) + ~ ( x ) g i v e s us F X ( x ) _> (X - )~0)F(x) + F x 0 ( x ). Hx(xo) = s

~

(~--~o)F(x)

+F~0(x)

t h e p o i n t x0, t h e

(~o < ~ )

Then

4- G(xo) - - F~(xo) -<~ ~.F(xo) + V(xo) - - (~, - - ~o)F(xo) - - F~fxo) = H~o(xo).

65

Since X and X0 w e r e c h o s e n a r b i t r a r i l y , it f o l l o w s f r o m the l a s t r e l a t i o n that Hx(x0) i s a n o n i n c r e a s ing f u n c t i o n of X. F i n a l l y , f r o m the i n e q u a l i t i e s (2.3) a n d (2.12) we o b t a i n the i n e q u a l i t y Hx(x0) < ~ when X >)~lSince ~ w a s c h o s e n a r b i t r a r i l y ,

we find t h a t H ~ ( x 0) ~ 0 a s X ~ 0.

2. We s p e c i f y s o m e l o c a l l y c o n v e x s p a c e X. We s h a l l c a l l t h e f u n c t i o n a l F : X ~ (R, ~o) s t r o n g l y c o n v e x o v e r L X, at t h e p o i n t x 0 i f t h e r e e x i s t s a c o n t i n u o u s h y p e r s u r f a e e f u n c t i o n a l f s u c h t h a t f(x0) = F(x0) a n d F(x) > f(x) f o r a l l x e X\{x0}. L e t L X, b e t h e s e t of c o n t i n u o u s h y p e r s u r f a c e f u n c t i o n a l s on X. T h e f u n c t i o n a l F : X ~ (R, oo), w h i c h i s s t r o n g l y c o n v e x o v e r L X, at t h e p o i n t x 0 E X w i l l be c a l l e d s t r o n g l y c o n v e x at x 0. A f u n c t i o n a l F : X ~ (R, oo) w i l l be c a l l e d s t r i c t l y ( s t r o n g l y ) c o n v e x if it i s s t r i c t l y ( s t r o n g l y ) c o n v e x a t e v e r y p o i n t of t h e s p a c e X. O b v i o u s l y f u n c t i o n a l s w h i c h a r e s t r i c t l y c o n v e x o v e r LX, at the p o i n t x 0 and f u n c t i o n a l s w h i c h a r e s t r o n g l y c o n v e x a t x 0 w i l l be c o n v e x o v e r L X, a t the p o i n t x 0 ; f u n c t i o n a l s w h i c h a r e s t r i c t l y c o n v e x o v e r LX Tand s t r o n g l y c o n v e x f u n c t i o n a l s w i l l be c o n v e x o v e r LX,. L E M M A 3. Let X be a r e f l e x i v e m e t r i z a b l e s p a c e . A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r the funct i o n a l F : X ~ (R, o~) to be s t r o n g l y c o n v e x i s t h a t it be s t r i c t l y c o n v e x o v e r LX,. P r o o f . The n e c e s s i t y of the c o n d i t i o n s of the l e m m a i s o b v i o u s . We s h a l l now p r o v e t h i e r s u f f i c i e n c y . S u p p o s e t h a t the f u n c t i o n a l F i s s t r i c t l y c o n v e x o v e r LX,. T h e n t h e r e e x i s t s a h y p e r s u r f a c e function f s u c h t h a t f(x 0 ) F ( x 0) a n d =

F(x) > f(x)

(2.13)

f o r a l l x E X\{x0}. L e t us c o n s i d e r t h e f u n c t i o n a l G(x) = F(x) - f(x). F r o m (2.13) it follows t h a t G(x 0) = 0, G(x) > 0 f o r a l l x E X\{x0}. A s s u m e t h a t the f u n c t i o n a l F is not s t r o n g l y c o n v e x at x0; t h e n t h e r e e x i s t s a n e i g h b o r h o o d V(x0) of the p o i n t x 0 w h i c h w i l l s a t i s f y t h e r e l a t i o n

(F(x)--l(x))~-

inf

inf

G(x)~---O,

(2.14)

We s h a l l d e n o t e b y Bp(x0) t h e s o l i d s p h e r e of r a d i u s p and c e n t e r x 0. L e t the s o l i d s p h e r e Bpl{x0) be s e l e c t e d in s u c h a w a y t h a t B2pl(X ) c V(x0). It f o l l o w s f r o m (2.14) t h a t t h e r e e x i s t s a s e q u e n c e of p o i n t s {Xm: x m e X\V(x0) } s u c h t h a t G(Xm) ~ 0 a s m ~ oo. We now c o n s t r u c t a s e q u e n c e of p o i n t s ~ym} a s f o l l o w s : y,~ = z~ t

o,

(x,, - xo).

(2.15)

0 (Xo, z~) F r o m B2pl(X 0)

c

V(x0) we h a v e

pJp(x o, x m)

< 1/2, and f r o m (2.15) i t f o l l o w s t h a t

t)(y,.,xo)=p(Zo, Zo)+Furthermore,

~)i O(Xo, X,,)

p(zo, X ~ ) = t),.

s i n c e the f u n c t i o n a l G is c o n v e x , we have t h e i n e q u a l i t y

( (i

)

pt

P'

O(Xo,X,,)

)G(xo)-]

(( P'

p(Xo, X~)

F r o m t h i s , s i n c e G(x) > 0 f o r a l l x E X and G(x m) ~ 0

G(xm)=

as m ~

P'

O(xo, x,~)

G(xm).

(2.16)

o~, we o b t a i n G(Ym) ~ 0 a s m ~ ~o.

Since X i s r e f l e x i v e and p(x 0, Ym) = Pl, we c a n s e l e c t f r o m the s e q u e n c e {Ym} a w e a k l y c o n v e r g e n t s u b s e q u e n c e {Ymk }. L e t Y0 be the l i m i t p o i n t of t h i s s u b s e q u e n c e . A p p l y i n g the c o r o l l a r y of T h e o r e m 1 to t h e f u n c t i o n a l G, we o b t a i n l i m G(Ymk) -< G(Y0), i . e . , G(y 0) = 0, w h i c h c o n t r a d i c t s t h e a s s u m p t i o n of s t r i c t m ~ o

c o n v e x i t y o v e r L X, at the p o i n t X 0 f o r the f u n c t i o n a l G, and h e n c e f o r the f u n c t i o n a l F . lemma.

T h i s p r o v e s the

L E M M A 4. L e t X be a l o c a l l y c o n v e x s p a c e and l e t Y b e a r e f l e x i v e m e t r i z a b l e s p a c e . S u p p o s e t h a t t h e r e e x i s t s a c o n t i n u o u s b i u n i q u e l i n e a r o p e r a t o r F : Y ~ X a n d t h a t t h e i m a g e of Y is e v e r y w h e r e d e n s e

66

i n X. A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r the f u n c t i o n a l F : X ~ be s t r i c t l y c o n v e x o v e r LX,.

(R, ~o) to b e s t r o n g l y c o n v e x is t h a t it

P r o o f . A s s u m e that the f u n c t i o n a l X i s not s t r o n g l y c o n v e x a n d t h a t at the s a m e t i m e it is s t r i c t l y c o n v e x o v e r LX,. T h e n t h e r e e x i s t s a c o n t i n u o u s h y p e r s u r f a c e f u n c t i o n a l 1 s u c h t h a t

F(xo) ---~ l(xo), f o r a l l x E X~{x0}.

F ( x ) > l(x)

(2.t7)

In t h a t c a s e , b y h y p o t h e s i s , t h e r e e x i s t s a n e i g h b o r h o o d V(x 0) of the p o i n t x 0 s u c h that in[

( ? ( z ) - - /(x) ) = 0.

(2.18)

Since l is a c o n t i n u o u s h y p e r s u r f a c e f u n c t i o n a l , it follows t h a t / ( x ) = g(x) + K , w h e r e g i s a 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 a n d K is a r e a l n u m b e r . By (2.18), t h e r e e x i s t s a s e q u e n c e of p o i n t s {Xm} s u c h t h a t x m E X ' ~ / ( x 0) for a l l m a n d

lira (F (x, 0 - - l(x,,) ) = O. Since the image of Y is evers~here

d e n s e in X, w e m a y a s s u m e

(2.19)

t h a t x m 6 F (Y) f o r a U v a l u e s o f me

L e t u s c o n s i d e r t h e f u n c t i o n a l s #~(y) = F ( F ( y ) ) a n d {(y) = g ( F ( y ) ) . S i n c e t h e l i n e a r o p e r a t o r F i s c o n t i n u o u s and b i u n l q u e , the f u n c t i o n a l ~ i s s e m i c o n t i n u o u s f r o m b e l o w a t the p o i n t Y0, w h i c h i s the p r e i m a g e of the p o i n t x0; the f u n c t i o n a l ~{ is c o n t i n u o u s a n d l i n e a r f o r the s a m e r e a s o n . F r o m (2.17) it follows t h a t

_~(y) =F(r(~)) >/g(r(~)) + K =

g(~) + K

f o r a l l y E Y. The e q u a l i t y h o l d s o n l y a t the point Y0. Suppose that {Ym} a r e t h e p r e i m a g e s of the p o i n t s {Xm} f o r a l l m . Since F is a c o n t i n u o u s o p e r a t o r , t h e r e e x i s t s a n e i g h b o r h o o d Vl(y0) of the p o i n t Y0 w h i c h w i l l not c o n t a i n t h e p o i n t s Ym f o r a l l m . F r o m (2.t9) it follows that lim (F(y~) - - s

K) :

m ---~

lira ( F ( F ( y , , ) ) - - g(F(y~) ) - - K) -----0. m - ~

Hence inf

( P ( y ) - - ~ ( y ) - - K) ~- 0.

(2.20)

H o w e v e r , b y (2.19), the f u n c t i o n a l F" i s s t r i c t l y c o n v e x o v e r L X, at the p o i n t Y0; b y L e m m a 3~ it is s t r o n g i y c o n v e x at Yo- T h i s c o n t r a d i c t s (2.20). T h e c o n t r a d i c t i o n we have j u s t a r r i v e d a t p r o v e s the s u f f i c i e n c y of the c o n d i t i o n s of the l e m m a . T h e i r n e c e s s i t y is o b v i o u s . 3. We s h a l l now p r o v e a n u m b e r of a s s e r t i o n s c o n c e r n i n g e x a c t m i n o r a n t s c o n v e x o v e r L X, a t the p o i n t x 0. Suppose t h a t X i s a l o c a l l y c o n v e x s p a c e , the f u n c t i o n a l K : X ~ (R, oo) i s c o n t i n u o u s a t the p o i n t x0, a n d the f u n c t i o n a l K has a n e x a c t m i n o r a n t c o n v e x o v e r L X, at the p o i n t x 0. T h e n the m i n o r a n t is s e m i e o n t i n u o u s f r o m b e l o w at the p o i n t x 0. To show t h i s , we note t h a t i n s u c h a c a s e the f u n c t i o n a l is b o u n d e d i n s o m e n e i g h b o r h o o d of the p o i n t x0, a n d t h e t a n g e n t f u n c t i o n a l to the e x a c t c o n v e x m i n o r a n t a t the p o i n t x 0 i s a l s o b o u n d e d i n t h i s n e i g h b o r h o o d . Since we a r e c o n s i d e r i n g o n l y b o r n o l o g i c a l s p a c e s , the t a n g e n t f u n c t i o n a l i s c o n t i n u o u s . F r o m t h i s it follows that the e x a c t m i n o r a n t c o n v e x at the p o i n t w i l l be a f u n c t i o n t h a t is c o n v e x o v e r LX, , a n d h e n c e it w i l l b e s e m i c o n t i n u o u s . Suppose that we a r e g i v e n a s e q u e n c e of f u n c t i o n a l s K m : X ~ (R, ~} (m = 1, 2 . . . . ) a n d a f u n c t i o n a l K 0 : X ~ (R, ~). We s h a l l s a y t h a t t h e f u n c t i o n a l {Kin} c o n v e r g e s u n i f o r m l y f r o m b e l o w to t h e f u n c t i o n a l K if f o r e v e r y e > 0 t h e r e e x i s t s a n m 0 s u c h t h a t for a l l m > m 0 we have / ; ~ (x) - K0(x) > - - 8.

LEMMA 5. Suppose that we a r e g i v e n a s e q u e n c e of f u n c t i o n a l s F m : X ~ (R, oo) (In = 1, 2 . . . . ) a n d a f u n c t i o n a l F 0 : X ~ (R, co) w h i c h i s s e m i c o n t i n u o u s f r o m below at s o m e p o i n t x 0 ~ X. Suppose t h a t F m ~ F 0 u n i f o r m l y f r o m below as m ~ ~o o n a n y b o u n d e d s e t of the s p a c e X. We a s s u m e t h a t t h e r e e x i s t s a f u n c t i o n a l 0 : X ~ (R, oo) s u c h that O ( x ) / p ( x , x0) ~ ~o a s p ( x , x0) ~ ~ ( h e r e x 0 i s s o m e p o i n t of X a n d the s p a c e X i s a s s u m e d to be m e t r i z a b l e ) . F u r t h e r m o r e , we a s s u m e that the f u n c t i o n a l 0 i s b o u n d e d f r o m b e t o w on a n y b o u n d e d s e t i n X. If Fm(X ) > 0(x) for a l l x a n d all m , t h e n the f u n c t i o n a l s F m w i l l h a v e e x a c t m i n e r a n t s

67

F m c o n v e x o v e r L X , a t t h e p o i n t x 0 f o r a l l m , a n d t h e s e q u e n c e of f u n c t i o n a l s {Fm} w i l l b e w e a k l y u n i f o r m ly convex. P r o o f . F i r s t of a l l we s h a l l p r o v e t h a t t h e f u n c t i o n a l 0 i s b o u n d e d b e l o w in t h e s p a c e X. To show t h i s , w e t a k e a n a r b i t r a r y r e a l n u m b e r N. F r o m t h e c o n d i t i o n i m p o s e d on t h e o r d e r o f m a g n i t u d e of the f u n c t i o n a l 0 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 s o l i d s p h e r e Bp(x0) s u c h t h a t 0(x) > N f o r a l l x ~ Bp(X0). A t t h e p o i n t s of t h e s o l i d s p h e r e Bp (x0) t h e f u n c t i o n a l 0 i s b o u n d e d , by h y p o t h e s i s . S u p p o s e t h a t 0(x) > M f o r a l l x e Bp (Xo). T h e n 0(x) > min(M, N)

(2.21)

f o r a l l x ~ X. F r o m t h i s i t f o l l o w s t h a t t h e f u n c t i o n a l 0 h a s an e x a c t m i n o r a n t c o n v e x o v e r L X, at t h e p o i n t x 0. L e t us d e n o t e t h i s m i n o r a n t b y ~. We s h a l l p r o v e t h a t ~ s a t i s f i e s t h e c o n d i t i o n s 0 (z) / ~ (x, x0) + co

as

a n d i s b o u n d e d on a n y s e t t h a t i s b o u n d e d in t h e s p a c e X.

0 (z, x.) - + co

T h e s e c o n d c o n d i t i o n f o l l o w s a t once f r o m (2.2D.

L e t us p r o v e t h e f i r s t c o n d i t i o n . To do t h i s , we c o n s i d e r t h e f u n c t i o n a l Kp(x) = Kp(X, x0). By v i r t u e of t h e c o n d i t i o n t h a t 0 ( x ) / p ( x , x 0) ~ ~o a s p(x, x 0) ~ .% t h e r e e x i s t s a s o l i d s p h e r e Bpt{X 0) s u c h t h a t (2.22)

0(x) > Kp(x)

for allx~Bpl(x0).

Let

min

0 ( x ) = M 1. We c o n s i d e r t h e f u n c t i o n a l Fl(x) = F(x) - K p l -

IMil.

S u p p o s e t h a t x E Bpl(X0); t h e n Ft (x) = Kp (x, xo) - - Kp, - - [M, I < M, < 0 (x) a n d i t f o l l o w s f r o m (2.22) t h a t Fl(x) < 0 (x) f o r a l l x E X. h a v e 0 (x) > F 1(x) f o r a l l x E X a n d 0 (x) / p (x, x0) - ~ co

as

F r o m t h i s , s i n c e t h e f u n c t i o n a l F 1 i s c o n v e x , we p (x, Zo) -+ co.

(2.23)

Now l e t us p r o v e the f o l l o w i n g : i f 1 is a c o n t i n u o u s h y p e r s u r f a c e f u n c t i o n a l s u c h t h a t F0(x) _> /(x) f o r a l l x E X, t h e n , s t a r t i n g f r o m s o m e m, we h a v e (2.24)

F , , ( x ) > l ( x ) - - e.

T o s h o w t h i s , w e n o t e t h a t a s a c o n s e q u e n c e of (2.23), t h e s e t on w h i c h 0(x) < /(x) i s b o u n d e d in the spaceX. We s h a l l d e n o t e t h i s s e t b y A . L e t m 0 b e s u c h t h a t m > m 0Fro(x) > F 0 ( x ) - e f o r a l l x E A . Then, f r o m t h i s a n d f r o m t h e f a c t t h a t Fm(X) > O (x) f o r a l l x E X and a l l m , we o b t a i n F,~(x) > l ( x ) - - e for all x ~ X.

(2.25)

S i n c e Fro(x) > 0(x), it f o l l o w s t h a t t h e f u n c t i o n a l s F m and F 0 h a v e e x a c t m i n o r a n t s c o n v e x o v e r LX, at t h e p o i n t x 0. We c o n s i d e r t h e f u n c t i o n a l F 0 . By t h e d e f i n i t i o n of the e x a c t m i n o r a n t c o n v e x o v e r LXr, a t t h e p o i n t x0, f o r a n y e > 0 t h e r e e x i s t s a c o n t i n u o u s h y p e r s u r f a c e f u n c t i o n a l l s u c h t h a t Fo(xo) - - l(xo) > e,

f o r a l l x ~ X.

Fo(x) ~

S i n c e , s t a r t i n g f r o m s o m e m, we have Fm(x) > / ( x ) - 2 a , F~ (x) > l(x) - - 2e.

Since e was chosen arbitrarily,

68

(2.27)

(2.28)

Now l e t us c h o o s e m to b e s u c h t h a t Fm (xo) < Fo (Xo) -k e

for allm

it follows that

we find f r o m (2.26) a n d (2.27) t h a t lira Fm (xo) ~>/~o (x0)

f o r alI x E X.

(2.26)

l(x)

> m i.

(2.29)

F r o m t h i s it follows that

~,~ (Zo) < ~'o (Xo) + f o r all x E X.

(2.30)

Since e was c h o s e n a r b i t r a r i l y , we find f r o m (2.28) a n d (2.30) t h a t lira Fra (xo) = lira _P,~(Xo) = Fo (xo)"

F r o m the l a s t r e l a t i o n a n d f r o m (2.27) it follows that the s e q u e n c e of f u n c t i o n a l s {Fro} is w e a k l y u n i f o r m l y c o n v e x at the p o i n t x 0 . Remark.

L e m m a 5 c a n be e x t e n d e d without m o d i f i c a t i o n to the c a s e of e x a c t m i n o r a n t s c o n v e x o v e r

LX,. 4. Let us c o n s i d e r in a l o c a l l y c o n v e x m e t r i z a b t e s p a c e a s e q u e n c e of f u n c t i o n a l s F r o : X ~ = 1, 2 , . . . ) which a r e c o n v e x o v e r L X, at the p o i n t x 0 and s a t i s f y the c o n d i t i o n s :

(R, ~ ) (m

a) F m ~ F 0 u n i f o r m l y on a n y b o u n d e d s e t in the s p a c e (the c o n d i t i o n of u n i f o r m c o n v e r g e n c e m a y be r e p l a c e d b y the c o n d i t i o n of u n i f o r m c o n v e r g e n c e f r o m below); b) the f u n c t i o n a l F 0 is s t r o n g l y c o n v e x at the p o i n t x 0 E X; c) t h e r e e x i s t s a f u n c t i o n a l 0 : X ~ (R, ~o) s u c h that 8(x)/p(x, x 0) ~ co a s p(x, x0) ~ t i o n a l 0 is b o u n d e d on a n y s e t t h a t i s b o u n d e d in the s p a c e X; F m ( x ) > 0 (x) f o r a l l x 6 X. Let us c o n s i d e r a l s o the s e q u e n c e of f u n c t i o n a l s Km : X ~ the following conditions:

from

d) the functional K 0 is semicontinuous from below below) on any set that is bounded in the space X; e) there exists a continuous

hypersurface

(R, ~o) a n d the f u n c t i o n a l K 0, w h i c h s a t i s f y

at the point x0, and K m

~

functional go and a real number

doF~(x) +go(x)
~*, a n d the f u n c -

K 0 uniformly

(uniformly

d o such that

doFo(x)+go(X)
T H E O R E M 3. Suppose that the f u n c t i o n a l s F m : X ~ ( R , ~ ) (m = 1, 2 . . . . ), F 0 : X ~ (R, ~); K m : X (R, ~o) (m = 1, 2 . . . . ), K 0 : X ~ (R, ~) s a t i s f y the above c o n d i t i o n s , {Xm} is a s e q u e n c e of p o i n t s of X, a n d x m ~ x 0 w e a k l y a s m ~ co. T h e n , if lira F m ( X m ) = F0(x0) , we have ~t3~OO

lira K~ (x,,) ~ K~(xo). m.-~m m

Proof. L e t us c o n s i d e r the f u n c t i o n a l s ?~Fm + Km. We s h a l l d e n o t e b y Gk t h e i r e x a c t m i n o r a n t s c o n v e x o v e r L X, at the p o i n t x 0. A n a l o g o u s l y , G~ is the e x a c t m i n o r a n t c o n v e x o v e r L X, for the f u n c t i o n a l XF o + K 0. The i n e q u a l i t y

)~Fm(x) -Jr Kin(x) > G~(x)

(2.31)

o b v i o u s l y holds for a l l m a n d a l l k, a n d the s a m e i s t r u e for the i n e q u a l i t y k F 0 (x) + K 0 (x) > G ~ (x) for all x~X. P a s s i n g to the l i m i t i n the i n e q u a l i t y (2.31), we o b t a i n lira ()~F,~ (xr~) q- Kr~ ( x m ) ) ~ lira Gxm (x,~).

By L e m m a 5, the s e q u e n c e of f u n c t i o n a l s { G f } Theorem 1 to this sequence, we obtain

(2.32)

i s w e a k l y u n i f o r m l y c o n v e x at the p o i n t X0o Applying

lim Gx'~ (xm) > C~0(xe). By h y p o t h e s i s , k F m ( X m ) ~ •F 0 (x0).

Comparing

the inequalities

(2.33) (2.31), (2.32), and (2.33)~ we obtain

r/l~'GO

)~Fo (xo) q- Ko (xo) -- G~~ (xo) > Ko (zo) -- lira K,~ (x,~).

69

T h e l e f t s i d e of t h i s i n e q u a l i t y a p p r o a c h e s 0 a s ~ ~ ~, b y L e m m a 2. _> K0(x0). T h i s p r o v e s t h e t h e o r e m .

H e n c e we find t h a t l i r a Km(Xm) ~Tl~ao

Now l e t us i m p o s e t h e f o l l o w i n g c o n d i t i o n s on the f u n c t i o n a l s {Kin}: d ' ) the f u n c t i o n a l K 0 i s c o n t i n u o u s a t t h e p o i n t x 0 and K m - - K 0 u n i f o r m l y on any s e t w h i c h i s b o u n d e d in t h e s p a c e ; e ' ) t h e r e e x i s t c o n t i n u o u s h y p e r s u r f a c e f u n c t i o n a l s go, l0 and r e a l n u m b e r s do, d 1 s u c h t h a t

doFf(x) +go(x) < K ~ ( x ) ;

doFo(x)-I-go(x) < K o ( x ) ;

d,F,~(x) +lo(X) >Kin(x);

d,F~

+Zo(X) >K0(x).

T H E O R E M 4. S u p p o s e t h a t t h e f u n c t i o n a l s {Fro}, {Kin}, F 0, K 0 s a t i s f y c o n d i t i o n s a ) - c ) , d ' ) , and e ' ) a n d t h a t {Xm} is a s e q u e n c e of p o i n t s of the s p a c e X w h i c h c o n v e r g e s w e a k l y to t h e p o i n t x 0. Then, if l i m Fm(Xm) = F0(x0), we h a v e m--~oo

lira K,. (x,.) ~ K~(xo). m - ~

To p r o v e t h i s , it i s s u f f i c i e n t to a p p l y T h e o r e m 3 to the s e q u e n c e s of f u n c t i o n a l s {Fm}, {Kin} a n d

{-Fm}, I-Kin}. C O R O L L A R Y . S u p p o s e t h a t F 0 i s a f u n c t i o n a l w h i c h i s s t r o n g l y c o n v e x at x 0 and K 0 s a t i s f i e s c o n d i t i o n s d') and e ' ) . T h e n , if {Xm} i s a s e q u e n c e o f p o i n t s of the s e t w h i c h c o n v e r g e s w e a k l y to x 0 and l i r a F 0 (Xm) = F0(x0), we h a v e ?n~oo

lira KQ(x~) ~ Ko(xo), m . ~

a n d i f K 0 s a t i s f i e s c o n d i t i o n d) a n d e), w e h a v e lira Km (xm) : Ko(xo). ~l--+ao

R e m a r k . In the c a s e of r e f l e x i v e s p a c e s o r s p a c e s w h i c h s a t i s f y the c o n d i t i o n s of L e m m a 4, s t r o n g c o n v e x i t y of t h e f u n c t i o n a l m a y b e r e p l a c e d b y t h e c o n d i t i o n of s t r i c t c o n v e x i t y o v e r L X , .

LITERATURE io

2.

70

CITED

Yu. G. Reshetnyak, "General theorems on semicontinuity and convergence with a functional," Sibirsk. Matem. Zh., 6, No. 5, 1051-1069 (1967). N. Dunford and J. T. Schwartz, Linear Operators [Russian translation], Vol. I, Izd. Inostr. Lit. (1962).