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Convergence Theorems for Best Approximations in a Nonreflexive Banach Space
Journal of Approximation Theory AT3170 Journal of Approximation Theory 93, 480490 (1998) Article No. AT983170
Convergence Theorems for Best Approximations in a Nonreflexive Banach Space Jian-Hua Wang Department of Mathematics, Anhui Normal University, Wuhu, Anhui, 241000, People's Republic of China Communicated by Gunther Nurnberger Received July 16, 1996; accepted in revised form May 1, 1997
In this paper, we study continuity properties of the mapping P: (x, A) P A (x) in a nonreflexive Banach space where P A is the metric projection onto A. Our results extend the existing convergence theorems on the best approximations in a reflexive Banach space to nonreflexive Banach spaces by using Wijsman convergence of sets. 1998 Academic Press
1. INTRODUCTION The problem of continuity for the mapping A P A (x) was first considered by Brosowski, Deutsch, and Nurnberger [2]. They considered a family [A t ] t # T of subsets of a normed linear space X parametrized by a topological space T and studied the continuity of t P At (x). In 1984, Tsukada [12] discussed the above problem but with a nonparametrized method. He proved that if the Banach space X is reflexive strictly convex, then when the closed convex sets [A n ] converge to A in the Mosco sense it is implied that [P An (x)] weakly converges to P A (x) for each x # X. Moreover if X has the property (H), the convergence is in the norm. Papageorgion and Kandilakis [8] generalized the work of Tsukada and studied the convergence of =-approximations. Recently, the author and Nan [14] gave a further description on the convergence of best approximations. In these convergence theorems for best approximations, one standard condition is that the Banach space X is reflexive. The purpose of this paper is to establish new convergence results for best approximations in a nonreflexive Banach space. For this purpose, we need a notion of strong Wijsman convergence for the sequence of nonempty sets. Under the assumption of the strong Wijsman convergence we give two continuity 480 0021-904598 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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theorems of the mapping P: (x, A) P A (x) in nonreflexive spaces. Moreover we prove that in a reflexive Banach space, if the sequence of closed convex subsets [A n ] converges to A in the Mosco sense, then [A n ] converges to A in the strong Wijsman sense. As a consequence, our results are natural extensions of the existing results on the continuity of metric projections P A (x) (as a set-valued mapping of (x, A)) in a reflexive Banach space [1216] to nonreflexive Banach spaces.
2. NOTATIONS AND DEFINITIONS Let [A n ] be a sequence of nonempty subsets of a Banach space. Define the weak limit superior of the sequence [A n ] to be the set w-lim A n =[x # X : x=w-lim x nk , x nk # A nk , k 1] n
n
and the strong limit inferior of [A n ] to be the set s- A n =[x # X : x=lim x n , x n # A n , n 1]. n
n
A sequence of nonempty subsets [A n ] of a Banach space X is said to converge to a set A in the Mosco sense if s- A n =w-lim A n =A. We will n n M write lim A. M A n =A or A n w n A sequence of nonempty subsets [A n ] of a Banach space X is said to converge to a set A in the Wijsman sense if lim d(x, A n )=d(x, A) for each n x in X, where d(x, A)=inf y # A &x& y&. We will write lim W A n =A or n W A. An w Let X be a Banach space and A be a subset of X. P A (x) denotes the set of all best approximations to x from A; i.e., P A (x)=[ y # A : &x& y&= d(x, A)]. The set A is proximinal (resp. Chebyshev) if P A (x) contains at least (resp. exactly) one point for each x in X. For any =>0, we say that an element z in A is an =-approximation of x in A if &x&z& d(x, A)+=. We will denote the set of all =-approximations by P A= (x). Let U(X ) denote the closed unit ball of a Banach space X and let S(X ) denote the unit sphere of X. Also co(B) means the convex hull of B. A sequence of nonempty subsets [A n ] of a Banach space X is said to converge to a set A in the strong Wijsman sense, if for each fixed x in X and any = n 0 + as n , we have lim &x&u n &=d(x, A) whenever u n # n =k sW A. co( k=n P Ak(x)) for n 1. We will write s-lim W A n =A or A n w n
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Clearly, [An ] converges to A in the Wijsman sense if and only if for each x in X and any = n 0 + as n , we have lim &x&a n &=d(x, A) whenever n = an # P Ann(x) for n 1. Furthermore, s-lim W A n =A implies limW A n =A. n n Let C be the set of all proximinal subsets of X. The mapping x P A (x) is said to be (weakly) upper semicontinuous if for each fixed x in X and any (weakly) open set W#P A (x), there exists a neighborhood U of x such that P A (U)/W. We say that the mapping P: (x, A) P A (x) is (weakly) upper semicontinuous at (x, A) in the strong Wijsman sense if, given a (weakly) open set W containing PA (x), any x n x, and any sequence [A n ]/C with sW A, there exists a corresponding integer N such that P An (x n )/W for An w all n N. The mapping P: (x, A) P A (x) is called (weakly) upper semicontinuous in the strong Wijsman sense if it is (weakly) upper semicontinuous at each (x, A) in the strong Wijsman sense. In [13], a Banach space X is said to have the proper (C-I) (resp. (C-II) or (C-III)), if x # S(X), [x n ] in U(X) with the property that for each $>0 there exists an integer N($) such that co([x] _ [x n : n N($)]) & (1&$) U(X)=<, then lim &x n &x&=0 (resp. [x n ] is relatively compact or [x n ] is relatively n weakly compact). We say that a Banach space X has the property (H), if for sequences on the unit sphere of X weak convergence is equivalent to norm convergence. We also mention some convexity properties for X. These definitions can be found in [4, 13, 15]. A Banach space X is locally nearly uniformly convex (LNUC) if for each x # S(X) and each =>0, there is a positive constant $=$(x, =) such that for all sequences [x n ] in U(X) with inf [&x n &x m &: n{m]>=, we have co([x] _ [x n : n 1]) & (1&$) U(X){<. A Banach space X is compactly locally fully k rotund (CL-kR), if for any sequence [x n ] in U(X) and for any x # S(X) with lim
n1, ..., nk
&x+x n1 + } } } +x nk &=k+1,
then [x n ] is relatively compact. Replacing the phrase ``[x n ] is relatively compact'' by ``[x n ] is relatively weakly compact,'' X is called a (WCL-kR) space. Finally, we give some results concerned with the strong Wijsman convergence of sets and the properties (C-II) and (C-III). They will be used in Section 3.
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CONVERGENCE THEOREMS
Proposition 2.1. Let X be a reflexive Banach space. If An and A are nonempty closed convex subsets of X, then limM A n =A implies s-limW A n =A. n
n
Proof. We prove the proposition by contradiction. Assume the contrary, =k that there exists an x 0 and sequences [= n ], [u ni ] with u ni # co( k=ni P Ak(x 0 )) such that = n 0 + as n , but | &x 0 &u ni &&d(x 0 , A)| >'>0 for i 1. Let =k (i) (i) (i) mi (i) (i) i uni = m j=1 * j a j where * j 0, j=1 * j =1, a j # P Ak(x 0 ) for some k n i , j=1, 2, ..., m i , i 1. Then mi
&x 0 &u ni & : * j(i) &x 0 &a j(i) & j=1
sup [d(x 0 , A k )+= k ]
for all i.
k ni
A n =A It is known from [1] that for a reflexive Banach space, lim n M : j=1, 2, ..., m , i 1] are implies lim W A n =A. It follows that [u ni ] and [a (i) i j n bounded and lim &x 0 &u ni & d(x 0 , A). Thus we can select a subsequence, i
denoted by [u ni ] again, such that lim &x 0 &u ni &<#
By the assumption that X is reflexive, [u ni ] has a weakly convergent subsequence. Without loss of generality, we may assume w-lim u ni = y. If y # A, i
then d(x 0 , A) &x 0 & y& &x 0 &u ni &<#
Therefore y A. By the strong separation theorem, there is an x * 0 # X* such that x* 0 ( y)>:>sup x* 0 (z). z#A
Since y=w-lim u ni , we can assume that x* 0 (u ni )>: for all i. Since u ni = i (i) (i) (i) i * a , there exists an index j , 1 j i m i , such that x* m i 0 (a ji )>:. j=1 j j (i) (i) Observe that [a ji ] is bounded. Hence [a ji ] has a weakly convergent subsequence; for convenience, assume z 0 =w-lim a j(i) . Then x* 0 (z 0 )= i i (i) (a ) :. However, by the assumption that lim A =A, we obtain lim x* 0 n M ji n
i
z0 # w-lim A n =A. Consequently, x* 0 (z 0 ) sup z # A x* 0 (z)<:. This is a conn tradiction. Therefore s-limW A n =A. n
Proposition 2.2. If [A n ] is an increasing sequence of nonempty convex subsets of a Banach space X, then s-lim A n = n=1 A n . Moreover, n W limM An = n=1 A n . n
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Proof. Let x # X. We first prove that lim d(x, A n )=d(x, n=1 A n ). Since n [A n ] is increasing, obviously lim d(x, A n ) d(x, n=1 A n ). On the other n hand, if y # A , we can choose yk # n n=1 n=1 A n such that &y& y k &<1k for k 1. Since A 1 /A 2 /..., we have y k # A n for all n large enough. Thus, &x& y& &x& y k &&&y& y k & d(x, A n )&1k for all n large enough and all k, which implies that
\
+
d x, . A n lim d(x, A n ). n=1
n
Therefore lim d(x, A n )=d(x, n=1 A n ). n + Now we prove s-lim W A n = as n and let n=1 A n . Let x # X, = n 0 n =k un # co( k=n P Ak(x)), n 1. As in the proof of Proposition 2.1, we can derive that &x&u n & sup[d(x, A k )+= k ]
(1)
k n
for all n. Since [A n ] is an increasing sequence of convex sets, clearly co( k=n A k )= k=n A k . Hence u n # A k for all k large enough. This means that &x&u n & d(x, A k )
(2)
for all k large enough. From (1), (2), and lim d(x, A n )=d(x, n=1 A n ), it n follows that lim &x&u n &=d(x, n=1 A n ). So s-lim W A n = n=1 A n . n
n
By using the fact that lim d(x, A n )=d(x, n=1 A n ) and n=1 A n is n convex, it is easy to verify that w-lim A n / n=1 A n /s- A n . Hence n n limM An = n=1 A n . n In [13], the following statements have been proved: (i) Every (LNUC) or (CL-kR) Banach space has the property (C-II); (ii) Every (WCL-kR) space has the property (C-III). For a reflexive Banach space, we have the following proposition about the property (C-II).
Proposition 2.3. Let X be a reflexive Banach space and let X have the property (H). Then X has the property (C-II). Proof. Let x # S(X) and [x n ]/U(X) be such that for each $>0, there exists an integer N($) satisfying co([x] _ [x n : n N($)]) & (1&$) U(X)=<.
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Since X is reflexive, [x n ] is relatively weakly compact. Suppose that [x ni ] is a weakly convergent subsequence of [x n ] and w-lim x ni =x 0 . It is known i that any closed convex set is weakly closed. So x 0 # co([x] _ [x n : n N($)]). Hence &x 0 & 1&$ for each $>0. Therefore we have &x 0 &=1. Because X has the property (H), it follows that x ni x 0 . This proves that [x n ] is relatively compact.
3. CONVERGENCE OF BEST APPROXIMATIONS IN NONREFLEXIVE BANACH SPACES Let X be a Banach space and X* be the dual space of X. Lemma 3.1. Let X be a Banach space with the property (C-II), A n /X (n 1), A in C, and s-limW A n =A. If x n x and y n # P An(x n ) (n 1), then n [ y n ] is relatively compact and its convergent subsequence converges to an element of P A(x). Proof. Let x # X, x n x, y n # P An(x n ), n=1, 2, ... . Clearly, d(x, A n ) &x& yn & &x&x n & + &x n & y n & = &x&x n & + d(x n , A n ) &x&x n & + |d(x n , A n )&d(x, A n )| +d(x, A n ). By [10] or [12, Remark, 302], for any nonempty subset A of X, d( } , A) is uniformly continuous and |d(x, A)& d( y, A)| &x& y& for any x, y # X. So we obtain that d(x, A n ) &x& y n & 2 &x&x n &+d(x, A n )
(3)
for all n. Since s-limW A n =A implies limW A n =A, we derive from (3) that n
n
lim &x& y n &=d(x, A).
(4)
n
Fix z # P A(x). Since limW A n =A, so lim d(z, A n )=d(z, A)=0. Thus there n n exists a sequence z n # A n , n 1, such that lim &z n &z&=0. Let = n = n =k 2 &x&x n &+&z&z n &+|d(x, A n )&d(x, A)|, and : n =d(x, co( k=n P Ak(x))). By (3), &x& y n & d(x, A n )+= n . This, along with y n # An , implies yn # P A=nn(x). Therefore : n &x& y n &. Since [: n ] is increasing, by (4) we =k obtain lim : n d(x, A). For each n, select u n # co( k=n P Ak(x)) with n =k &x&u n &
lim : n =d(x, A). n
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(5)
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Observe that &x & z n & &x & z& + &z & z n & &z & z n & + |d(x, A n ) & d(x, A)| +d(x, A n ) d(x, A n )+= n for all n. Therefore z n # P A=nn(x) for n 1. Suppose d(x, A){0. Given $>0, in view of (5) and lim &z&z zn &=0, there n exists an integer N1($) such that when n N 1($), we have &z&z n & d(x, A) } $4.
: n d(x, A)(1&$4),
Further, if *i 0, i=0, 1, ..., m, m i=0 * i =1, and N 1($) n 0 n 1 } } } n m , we get that &x&(* 0 z+* 1 y n1 + } } } +* m y nm )& &x&(* 0 z n0 +* 1 y n1 + } } } + * m y nm )&&* 0 &z&z n0 & : n0 &* 0 &z&z n0 & d(x, A)(1&$2). Thus,
"
*0
x& y nm x&z 1&$2. + } } } +* m d(x, A) d(x, A)
"
d(x, A n )=d(x, A) and (4), there exists an integer N($) ( N 1($)) By lim n such that 1
1
} d(x, A) &d(x, A )+2 &x&x & } sup &x& y &<$2 k
n
n
k
for n N($). Thus if N($) n 1 } } } n m , we get x&z
x& y n1
" * d(x, A)+* d(x, A )+2 &x&x 0
1
n1
+ } } } +* m
"
*0
n1
&
x& y nm d(x, A nm )+2 &x&x nm &
"
x& y n1 x& y nm x&z +* 1 + } } } +* m d(x, A) d(x, A) d(X, A)
m
& : *i i=1
"
1
1
} d(x, A) &d(x, A )+2 &x&x & } &x& y & ni
ni
ni
>1&$2&$2=1&$. This shows that for each fixed $>0, there exists an integer N($) such that co
x&z
x& y n
\{d(x, A)= _ {d(x, A )+2 &x&x & : n N($)=+ n
n
& (1&$) U(X)=<.
(6)
By (3) and the assumption that X has the property (C-II), we obtain that [(x& y n )(d(x, A n )+2 &x&x n &)] is relatively compact. Further [ y n ] is
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relatively compact. If d(x, A)=0, from (4) it follows that [ y n ] is a convergent sequence. Of course it is relatively compact. Let y ni y. Observe that d( y, A ni ) &y& y ni & 0 and d( y, A)=lim d( y, A ni ), so d( y, A)=0. i
Since any proximinal set is closed, hence y # A. By (4) we have &x& y&= lim &x& y ni &=d(x, A); thus y # P A(x). i
Theorem 3.1. Let X be a Banach space with the property (C-II), and A # C. Then the mapping P: (x, A) P A(x) is upper semicontinuous in the strong Wijsman sense. Proof. Assume the contrary, that Theorem 3.1 is not true. Then for some (x 0 , A 0 ) and a norm open set W 0 #P A0(x 0 ) and corresponding convergent sequences x n x 0 , s-lim W A n =A 0 , we can choose a subsequence n [n i ] of [n] such that P An (x ni ) /3 W 0 for all i. Let y ni # P An (x ni )"W 0 , i 1. i i By Lemma 3.1 it follows that [ y ni ] has a subsequence which converges to an element of P A0(x 0 ). This means that there exists y ni # W 0 for some i large enough. This is impossible. This completes the proof of Theorem 3.1. An immediate consequence of Theorem 3.1 is the following corollary. Corollary 3.1 [13]. Let X be a Banach space with the property (C-II) and A be a proximinal convex subset in X. Then the mapping x P A(x) is upper semicontinuous. Using Theorem 3.1 and Propositions 2.1 and 2.3 we derive the following corollaries. Corollary 3.2 [14]. Let X be a reflexive Banach space with the property (H). If A n (n 1) and A are closed convex subsets of X with lim M n A n =A, then for each x in X and all [x n ] we have lim(sup y # PA (xn ) n n d( y, P A(x)))=0 whenever x n x. Corollary 3.3 [12]. Let X be a strictly convex reflexive Banach space with the property (H). If A n (n 1) and A are closed convex subsets of X with lim M A n =A, then lim &P An(x)&P A(x)&=0 for all x in X. n
n
Lemma 3.2. Let X be a Banach space with the property (C-III), A n /X (n 1), A in C and s-lim A n =A. If x n x and y n # P An(x n ) (n 1) then n W [ y n ] is relatively weakly compact. If in addition lim M A n =A then the n weakly convergent subsequence of [ y n ] converges to an element of P A(x) weakly.
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Proof. The proof of the first part is the same as Lemma 3.1. Now we prove the second part. Let [ y ni ] be a weakly convergent subsequence of [ y n ] and w-lim y ni = y. Since lim M A n =A, so y # A. By the weak lower i n semicontinuity of the norm and (4) we can write that &x& y& &x& y ni &=d(x, A). i
But recall that y # A. So y # P A(x). A result of Papageorgion and Kandilakis [8, Proposition 2.2] tells us that in a general Banach space, if lim M A n =A then w-lim P An(x)/P A(x) n
n
for all x in X. Their proof also applies to the following case: If limM A n =A n and x n x, then w-lim P An(x n )/P A(x). Further, the second part of n Lemma 3.2 can be derived immediately by Proposition 2.2 of [8]. sW&M If s-lim W A n =lim M A n =A, we will write A n ww A. Following weak n n upper semicontinuity in the strong Wijsman sense, we can similarly introduce the concept of weak upper semicontinuity in the strong Wijsman Mosco sense for the mapping P: (x, A) P A(x).
Theorem 3.2. Let X be a Banach space with the property (C-III), and A in C. Then the mapping P: (x, A) P A (x) is weakly upper semicontinuous in the strong WijsmanMosco sense. If, in addition X is strictly convex and A is convex, then the mapping P: (x, A) P A (x) is weakly continuous in the strong Wijsman sense. Proof. The weak upper semicontinuity of P in the strong Wijsman Mosco sense can be proved by Lemma 3.2. The proof proceeds as in Theorem 3.1. Next we prove the weak continuity of P in the strong Wijsman sense. Let u # S(X) and [u n ] in U(X ) with the property that for each $>0 there exists an integer N($) such that co([u] _ [u n : n N($)]) & (1&$) U(X )=<. We will prove w-lim u n =u. Since X has the property (C-III), [u n ] has a n weakly convergent subsequence. Let w-lim u ni =u 0 . Since u 0 is in i
co([u n : n N($)]), so 12 (u+u 0 ) # co([u] _ [u n : n N($)]) for every $>0. Therefore we obtain that u 0 # S(X) and & 12 (u+u 0 )&=1. From the strict convexity of X it follows u=u 0 ; thus, w-lim u n =u. n Now let x n x and s-lim W A n =A, where A n (n 1) and A are n proximinal convex subsets of X. If y n # P An (x n ) for every n and z # P A (x), from the proof of Lemma 3.1 we know that when x A, then for each
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fixed $>0 there is an integer N($) such that co([(x&z)d(x, A)] _ [(x& y n )(d(x, A n )+2 &x&x n &) : n N($)]) & (1&$) U(X )=<. Consequently, w-lim y n =z # P A (x). If x # A, by (4) in Lemma 3.1 we have n y n x # P A (x). Observe that if X is strictly convex and A is a proximinal convex subset, then A is a Chebyshev set. Therefore w-lim P An (x n )=P A (x). n Recall that a reflexive Banach space has the property (C-III) and lim M A n =A implies s-lim W A n =A. Thus we have the following corollaries n n of Theorem 3.2. Corollary 3.4 [12]. Let X be a reflexive and strictly convex Banach space, A n (n 1) and let A be closed convex subsets of X with lim M A n =A. n Then w-lim P An (x)=P A (x) for each x in X. n
Corollary 3.5 [13]. Let X be a Banach space with the property (C-III) and A be a proximinal convex subset in X. Then the mapping x P A (x) is weakly upper semicontinuous. Remark 3.1. Observe that a Banach space X has the property (C-I) if and only if X has the property (C-II) and X is strictly convex. By Theorem 3.1 it is clear that if X has the property (C-I), A n (n 1) and A are proximinal convex sets with s-lim W A n =A, then for each x in X, n lim &P An (x n )&P A (x)&=0 whenever x n x. n
Remark 3.2. Theorem 3.1 is true for (LNUC) or (CL-kR) and Theorem 3.2 is true for (WCL-kR). Remark 3.3. In [11], F. Sullivan introduced the locally k uniformly rotund (Lk-UR) spaces. He proved that if M is a Chebyshev subspace of the (L2-UR) space, then the mapping x P M (x) is norm continuous. In 1985, Yu Xintai [16] extended Sullivan's theorem [11] to (Lk-UR) spaces. From Theorem 1 of [6] it is known that every (Lk-UR) space is (CL-kR). According to Remark 3.2 we get that Theorem 3.1 is true replacing the property (C-II) by (Lk-UR). Consequently, our results generalize the continuity theorems of [11, 16]. REFERENCES 1. J. M. Borwein and S. Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), 843851. 2. B. Brosowski, F. Deutsch, and G. Nurnberger, Parametric approximation, J. Approx. Theory 29 (1980), 261271. 3. J. Diestel, ``Geometry of Banach Spaces-Selected Topics,'' Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, BerlinHeidelbergNew York, 1975. 4. D. Kutzarova and Bor-Luh Lin, Locally k-nearly uniformly convex Banach spaces, Math. Balkanica 8, No. 23 (1994), 203210.
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5. Na Qiyuan, On fully convex and locally fully convex Banach space, Acta Math. Sci. 10, No. 3 (1990), 327343. 6. Nan Chao-Xun and Wang Jian-Hua, On the Lk-UR and L-kR spaces, Math. Proc. Cambridge Philos. Soc. 104 (1988), 521526. 7. B. B. Panda and O. P. Kapoor, A generalization of local uniform convexity of the norm, J. Math. Anal. Appl. 52 (1975), 300308. 8. N. S. Papageorgion and D. A. Kandilakis, Convergence in approximation and nonsmooth analysis, J. Approx. Theory 49 (1987), 4154. 9. J. Rainwater, Local uniform convexity of Day's norm on c 0 (1 ), Proc. Amer. Math. Soc. 22 (1969), 335339. 10. I. Singer, ``Best Approximation in Normed-Linear Spaces by Elements of Linear Subspaces,'' Springer-Verlag, Berlin, 1973. 11. F. Sullivan, A generalization of uniformly rotund Banach spaces, Canad. J. Math. 31 (1979), 628646. 12. M. Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), 301309. 13. Wang Jianhua, Some results on the continuity of metric projections, Math. Appl. 8, No. 1 (1995), 8084. 14. Wang Jianhua and Nan Chao-Xun, On the convergence of =-approximation, in ``Banach Space Theory and Its Application,'' Wuhan Univ. Press, 1996. 15. Wang Jianhua and Wang Musan, Compactly locally fully convex spaces, Kexue Tongbao 36 (1991), 796. 16. Yu Xintai, On LKUR spaces, Chinese Ann. Math. (Ser. B) 6, No. 4 (1985), 465469.
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Convergence Theorems for Best Approximations in a Nonreflexive Banach Space Jian-Hua Wang Department of Mathematics, Anhui Normal University, Wuhu, Anhui, 241000, People's Republic of China Communicated by Gunther Nurnberger Received July 16, 1996; accepted in revised form May 1, 1997
In this paper, we study continuity properties of the mapping P: (x, A) P A (x) in a nonreflexive Banach space where P A is the metric projection onto A. Our results extend the existing convergence theorems on the best approximations in a reflexive Banach space to nonreflexive Banach spaces by using Wijsman convergence of sets. 1998 Academic Press
1. INTRODUCTION The problem of continuity for the mapping A P A (x) was first considered by Brosowski, Deutsch, and Nurnberger [2]. They considered a family [A t ] t # T of subsets of a normed linear space X parametrized by a topological space T and studied the continuity of t P At (x). In 1984, Tsukada [12] discussed the above problem but with a nonparametrized method. He proved that if the Banach space X is reflexive strictly convex, then when the closed convex sets [A n ] converge to A in the Mosco sense it is implied that [P An (x)] weakly converges to P A (x) for each x # X. Moreover if X has the property (H), the convergence is in the norm. Papageorgion and Kandilakis [8] generalized the work of Tsukada and studied the convergence of =-approximations. Recently, the author and Nan [14] gave a further description on the convergence of best approximations. In these convergence theorems for best approximations, one standard condition is that the Banach space X is reflexive. The purpose of this paper is to establish new convergence results for best approximations in a nonreflexive Banach space. For this purpose, we need a notion of strong Wijsman convergence for the sequence of nonempty sets. Under the assumption of the strong Wijsman convergence we give two continuity 480 0021-904598 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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481
theorems of the mapping P: (x, A) P A (x) in nonreflexive spaces. Moreover we prove that in a reflexive Banach space, if the sequence of closed convex subsets [A n ] converges to A in the Mosco sense, then [A n ] converges to A in the strong Wijsman sense. As a consequence, our results are natural extensions of the existing results on the continuity of metric projections P A (x) (as a set-valued mapping of (x, A)) in a reflexive Banach space [1216] to nonreflexive Banach spaces.
2. NOTATIONS AND DEFINITIONS Let [A n ] be a sequence of nonempty subsets of a Banach space. Define the weak limit superior of the sequence [A n ] to be the set w-lim A n =[x # X : x=w-lim x nk , x nk # A nk , k 1] n
n
and the strong limit inferior of [A n ] to be the set s- A n =[x # X : x=lim x n , x n # A n , n 1]. n
n
A sequence of nonempty subsets [A n ] of a Banach space X is said to converge to a set A in the Mosco sense if s- A n =w-lim A n =A. We will n n M write lim A. M A n =A or A n w n A sequence of nonempty subsets [A n ] of a Banach space X is said to converge to a set A in the Wijsman sense if lim d(x, A n )=d(x, A) for each n x in X, where d(x, A)=inf y # A &x& y&. We will write lim W A n =A or n W A. An w Let X be a Banach space and A be a subset of X. P A (x) denotes the set of all best approximations to x from A; i.e., P A (x)=[ y # A : &x& y&= d(x, A)]. The set A is proximinal (resp. Chebyshev) if P A (x) contains at least (resp. exactly) one point for each x in X. For any =>0, we say that an element z in A is an =-approximation of x in A if &x&z& d(x, A)+=. We will denote the set of all =-approximations by P A= (x). Let U(X ) denote the closed unit ball of a Banach space X and let S(X ) denote the unit sphere of X. Also co(B) means the convex hull of B. A sequence of nonempty subsets [A n ] of a Banach space X is said to converge to a set A in the strong Wijsman sense, if for each fixed x in X and any = n 0 + as n , we have lim &x&u n &=d(x, A) whenever u n # n =k sW A. co( k=n P Ak(x)) for n 1. We will write s-lim W A n =A or A n w n
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Clearly, [An ] converges to A in the Wijsman sense if and only if for each x in X and any = n 0 + as n , we have lim &x&a n &=d(x, A) whenever n = an # P Ann(x) for n 1. Furthermore, s-lim W A n =A implies limW A n =A. n n Let C be the set of all proximinal subsets of X. The mapping x P A (x) is said to be (weakly) upper semicontinuous if for each fixed x in X and any (weakly) open set W#P A (x), there exists a neighborhood U of x such that P A (U)/W. We say that the mapping P: (x, A) P A (x) is (weakly) upper semicontinuous at (x, A) in the strong Wijsman sense if, given a (weakly) open set W containing PA (x), any x n x, and any sequence [A n ]/C with sW A, there exists a corresponding integer N such that P An (x n )/W for An w all n N. The mapping P: (x, A) P A (x) is called (weakly) upper semicontinuous in the strong Wijsman sense if it is (weakly) upper semicontinuous at each (x, A) in the strong Wijsman sense. In [13], a Banach space X is said to have the proper (C-I) (resp. (C-II) or (C-III)), if x # S(X), [x n ] in U(X) with the property that for each $>0 there exists an integer N($) such that co([x] _ [x n : n N($)]) & (1&$) U(X)=<, then lim &x n &x&=0 (resp. [x n ] is relatively compact or [x n ] is relatively n weakly compact). We say that a Banach space X has the property (H), if for sequences on the unit sphere of X weak convergence is equivalent to norm convergence. We also mention some convexity properties for X. These definitions can be found in [4, 13, 15]. A Banach space X is locally nearly uniformly convex (LNUC) if for each x # S(X) and each =>0, there is a positive constant $=$(x, =) such that for all sequences [x n ] in U(X) with inf [&x n &x m &: n{m]>=, we have co([x] _ [x n : n 1]) & (1&$) U(X){<. A Banach space X is compactly locally fully k rotund (CL-kR), if for any sequence [x n ] in U(X) and for any x # S(X) with lim
n1, ..., nk
&x+x n1 + } } } +x nk &=k+1,
then [x n ] is relatively compact. Replacing the phrase ``[x n ] is relatively compact'' by ``[x n ] is relatively weakly compact,'' X is called a (WCL-kR) space. Finally, we give some results concerned with the strong Wijsman convergence of sets and the properties (C-II) and (C-III). They will be used in Section 3.
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CONVERGENCE THEOREMS
Proposition 2.1. Let X be a reflexive Banach space. If An and A are nonempty closed convex subsets of X, then limM A n =A implies s-limW A n =A. n
n
Proof. We prove the proposition by contradiction. Assume the contrary, =k that there exists an x 0 and sequences [= n ], [u ni ] with u ni # co( k=ni P Ak(x 0 )) such that = n 0 + as n , but | &x 0 &u ni &&d(x 0 , A)| >'>0 for i 1. Let =k (i) (i) (i) mi (i) (i) i uni = m j=1 * j a j where * j 0, j=1 * j =1, a j # P Ak(x 0 ) for some k n i , j=1, 2, ..., m i , i 1. Then mi
&x 0 &u ni & : * j(i) &x 0 &a j(i) & j=1
sup [d(x 0 , A k )+= k ]
for all i.
k ni
A n =A It is known from [1] that for a reflexive Banach space, lim n M : j=1, 2, ..., m , i 1] are implies lim W A n =A. It follows that [u ni ] and [a (i) i j n bounded and lim &x 0 &u ni & d(x 0 , A). Thus we can select a subsequence, i
denoted by [u ni ] again, such that lim &x 0 &u ni &<#
By the assumption that X is reflexive, [u ni ] has a weakly convergent subsequence. Without loss of generality, we may assume w-lim u ni = y. If y # A, i
then d(x 0 , A) &x 0 & y& &x 0 &u ni &<#
Therefore y A. By the strong separation theorem, there is an x * 0 # X* such that x* 0 ( y)>:>sup x* 0 (z). z#A
Since y=w-lim u ni , we can assume that x* 0 (u ni )>: for all i. Since u ni = i (i) (i) (i) i * a , there exists an index j , 1 j i m i , such that x* m i 0 (a ji )>:. j=1 j j (i) (i) Observe that [a ji ] is bounded. Hence [a ji ] has a weakly convergent subsequence; for convenience, assume z 0 =w-lim a j(i) . Then x* 0 (z 0 )= i i (i) (a ) :. However, by the assumption that lim A =A, we obtain lim x* 0 n M ji n
i
z0 # w-lim A n =A. Consequently, x* 0 (z 0 ) sup z # A x* 0 (z)<:. This is a conn tradiction. Therefore s-limW A n =A. n
Proposition 2.2. If [A n ] is an increasing sequence of nonempty convex subsets of a Banach space X, then s-lim A n = n=1 A n . Moreover, n W limM An = n=1 A n . n
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Proof. Let x # X. We first prove that lim d(x, A n )=d(x, n=1 A n ). Since n [A n ] is increasing, obviously lim d(x, A n ) d(x, n=1 A n ). On the other n hand, if y # A , we can choose yk # n n=1 n=1 A n such that &y& y k &<1k for k 1. Since A 1 /A 2 /..., we have y k # A n for all n large enough. Thus, &x& y& &x& y k &&&y& y k & d(x, A n )&1k for all n large enough and all k, which implies that
\
+
d x, . A n lim d(x, A n ). n=1
n
Therefore lim d(x, A n )=d(x, n=1 A n ). n + Now we prove s-lim W A n = as n and let n=1 A n . Let x # X, = n 0 n =k un # co( k=n P Ak(x)), n 1. As in the proof of Proposition 2.1, we can derive that &x&u n & sup[d(x, A k )+= k ]
(1)
k n
for all n. Since [A n ] is an increasing sequence of convex sets, clearly co( k=n A k )= k=n A k . Hence u n # A k for all k large enough. This means that &x&u n & d(x, A k )
(2)
for all k large enough. From (1), (2), and lim d(x, A n )=d(x, n=1 A n ), it n follows that lim &x&u n &=d(x, n=1 A n ). So s-lim W A n = n=1 A n . n
n
By using the fact that lim d(x, A n )=d(x, n=1 A n ) and n=1 A n is n convex, it is easy to verify that w-lim A n / n=1 A n /s- A n . Hence n n limM An = n=1 A n . n In [13], the following statements have been proved: (i) Every (LNUC) or (CL-kR) Banach space has the property (C-II); (ii) Every (WCL-kR) space has the property (C-III). For a reflexive Banach space, we have the following proposition about the property (C-II).
Proposition 2.3. Let X be a reflexive Banach space and let X have the property (H). Then X has the property (C-II). Proof. Let x # S(X) and [x n ]/U(X) be such that for each $>0, there exists an integer N($) satisfying co([x] _ [x n : n N($)]) & (1&$) U(X)=<.
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Since X is reflexive, [x n ] is relatively weakly compact. Suppose that [x ni ] is a weakly convergent subsequence of [x n ] and w-lim x ni =x 0 . It is known i that any closed convex set is weakly closed. So x 0 # co([x] _ [x n : n N($)]). Hence &x 0 & 1&$ for each $>0. Therefore we have &x 0 &=1. Because X has the property (H), it follows that x ni x 0 . This proves that [x n ] is relatively compact.
3. CONVERGENCE OF BEST APPROXIMATIONS IN NONREFLEXIVE BANACH SPACES Let X be a Banach space and X* be the dual space of X. Lemma 3.1. Let X be a Banach space with the property (C-II), A n /X (n 1), A in C, and s-limW A n =A. If x n x and y n # P An(x n ) (n 1), then n [ y n ] is relatively compact and its convergent subsequence converges to an element of P A(x). Proof. Let x # X, x n x, y n # P An(x n ), n=1, 2, ... . Clearly, d(x, A n ) &x& yn & &x&x n & + &x n & y n & = &x&x n & + d(x n , A n ) &x&x n & + |d(x n , A n )&d(x, A n )| +d(x, A n ). By [10] or [12, Remark, 302], for any nonempty subset A of X, d( } , A) is uniformly continuous and |d(x, A)& d( y, A)| &x& y& for any x, y # X. So we obtain that d(x, A n ) &x& y n & 2 &x&x n &+d(x, A n )
(3)
for all n. Since s-limW A n =A implies limW A n =A, we derive from (3) that n
n
lim &x& y n &=d(x, A).
(4)
n
Fix z # P A(x). Since limW A n =A, so lim d(z, A n )=d(z, A)=0. Thus there n n exists a sequence z n # A n , n 1, such that lim &z n &z&=0. Let = n = n =k 2 &x&x n &+&z&z n &+|d(x, A n )&d(x, A)|, and : n =d(x, co( k=n P Ak(x))). By (3), &x& y n & d(x, A n )+= n . This, along with y n # An , implies yn # P A=nn(x). Therefore : n &x& y n &. Since [: n ] is increasing, by (4) we =k obtain lim : n d(x, A). For each n, select u n # co( k=n P Ak(x)) with n =k &x&u n &
lim : n =d(x, A). n
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(5)
486
JIAN-HUA WANG
Observe that &x & z n & &x & z& + &z & z n & &z & z n & + |d(x, A n ) & d(x, A)| +d(x, A n ) d(x, A n )+= n for all n. Therefore z n # P A=nn(x) for n 1. Suppose d(x, A){0. Given $>0, in view of (5) and lim &z&z zn &=0, there n exists an integer N1($) such that when n N 1($), we have &z&z n & d(x, A) } $4.
: n d(x, A)(1&$4),
Further, if *i 0, i=0, 1, ..., m, m i=0 * i =1, and N 1($) n 0 n 1 } } } n m , we get that &x&(* 0 z+* 1 y n1 + } } } +* m y nm )& &x&(* 0 z n0 +* 1 y n1 + } } } + * m y nm )&&* 0 &z&z n0 & : n0 &* 0 &z&z n0 & d(x, A)(1&$2). Thus,
"
*0
x& y nm x&z 1&$2. + } } } +* m d(x, A) d(x, A)
"
d(x, A n )=d(x, A) and (4), there exists an integer N($) ( N 1($)) By lim n such that 1
1
} d(x, A) &d(x, A )+2 &x&x & } sup &x& y &<$2 k
n
n
k
for n N($). Thus if N($) n 1 } } } n m , we get x&z
x& y n1
" * d(x, A)+* d(x, A )+2 &x&x 0
1
n1
+ } } } +* m
"
*0
n1
&
x& y nm d(x, A nm )+2 &x&x nm &
"
x& y n1 x& y nm x&z +* 1 + } } } +* m d(x, A) d(x, A) d(X, A)
m
& : *i i=1
"
1
1
} d(x, A) &d(x, A )+2 &x&x & } &x& y & ni
ni
ni
>1&$2&$2=1&$. This shows that for each fixed $>0, there exists an integer N($) such that co
x&z
x& y n
\{d(x, A)= _ {d(x, A )+2 &x&x & : n N($)=+ n
n
& (1&$) U(X)=<.
(6)
By (3) and the assumption that X has the property (C-II), we obtain that [(x& y n )(d(x, A n )+2 &x&x n &)] is relatively compact. Further [ y n ] is
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487
relatively compact. If d(x, A)=0, from (4) it follows that [ y n ] is a convergent sequence. Of course it is relatively compact. Let y ni y. Observe that d( y, A ni ) &y& y ni & 0 and d( y, A)=lim d( y, A ni ), so d( y, A)=0. i
Since any proximinal set is closed, hence y # A. By (4) we have &x& y&= lim &x& y ni &=d(x, A); thus y # P A(x). i
Theorem 3.1. Let X be a Banach space with the property (C-II), and A # C. Then the mapping P: (x, A) P A(x) is upper semicontinuous in the strong Wijsman sense. Proof. Assume the contrary, that Theorem 3.1 is not true. Then for some (x 0 , A 0 ) and a norm open set W 0 #P A0(x 0 ) and corresponding convergent sequences x n x 0 , s-lim W A n =A 0 , we can choose a subsequence n [n i ] of [n] such that P An (x ni ) /3 W 0 for all i. Let y ni # P An (x ni )"W 0 , i 1. i i By Lemma 3.1 it follows that [ y ni ] has a subsequence which converges to an element of P A0(x 0 ). This means that there exists y ni # W 0 for some i large enough. This is impossible. This completes the proof of Theorem 3.1. An immediate consequence of Theorem 3.1 is the following corollary. Corollary 3.1 [13]. Let X be a Banach space with the property (C-II) and A be a proximinal convex subset in X. Then the mapping x P A(x) is upper semicontinuous. Using Theorem 3.1 and Propositions 2.1 and 2.3 we derive the following corollaries. Corollary 3.2 [14]. Let X be a reflexive Banach space with the property (H). If A n (n 1) and A are closed convex subsets of X with lim M n A n =A, then for each x in X and all [x n ] we have lim(sup y # PA (xn ) n n d( y, P A(x)))=0 whenever x n x. Corollary 3.3 [12]. Let X be a strictly convex reflexive Banach space with the property (H). If A n (n 1) and A are closed convex subsets of X with lim M A n =A, then lim &P An(x)&P A(x)&=0 for all x in X. n
n
Lemma 3.2. Let X be a Banach space with the property (C-III), A n /X (n 1), A in C and s-lim A n =A. If x n x and y n # P An(x n ) (n 1) then n W [ y n ] is relatively weakly compact. If in addition lim M A n =A then the n weakly convergent subsequence of [ y n ] converges to an element of P A(x) weakly.
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Proof. The proof of the first part is the same as Lemma 3.1. Now we prove the second part. Let [ y ni ] be a weakly convergent subsequence of [ y n ] and w-lim y ni = y. Since lim M A n =A, so y # A. By the weak lower i n semicontinuity of the norm and (4) we can write that &x& y& &x& y ni &=d(x, A). i
But recall that y # A. So y # P A(x). A result of Papageorgion and Kandilakis [8, Proposition 2.2] tells us that in a general Banach space, if lim M A n =A then w-lim P An(x)/P A(x) n
n
for all x in X. Their proof also applies to the following case: If limM A n =A n and x n x, then w-lim P An(x n )/P A(x). Further, the second part of n Lemma 3.2 can be derived immediately by Proposition 2.2 of [8]. sW&M If s-lim W A n =lim M A n =A, we will write A n ww A. Following weak n n upper semicontinuity in the strong Wijsman sense, we can similarly introduce the concept of weak upper semicontinuity in the strong Wijsman Mosco sense for the mapping P: (x, A) P A(x).
Theorem 3.2. Let X be a Banach space with the property (C-III), and A in C. Then the mapping P: (x, A) P A (x) is weakly upper semicontinuous in the strong WijsmanMosco sense. If, in addition X is strictly convex and A is convex, then the mapping P: (x, A) P A (x) is weakly continuous in the strong Wijsman sense. Proof. The weak upper semicontinuity of P in the strong Wijsman Mosco sense can be proved by Lemma 3.2. The proof proceeds as in Theorem 3.1. Next we prove the weak continuity of P in the strong Wijsman sense. Let u # S(X) and [u n ] in U(X ) with the property that for each $>0 there exists an integer N($) such that co([u] _ [u n : n N($)]) & (1&$) U(X )=<. We will prove w-lim u n =u. Since X has the property (C-III), [u n ] has a n weakly convergent subsequence. Let w-lim u ni =u 0 . Since u 0 is in i
co([u n : n N($)]), so 12 (u+u 0 ) # co([u] _ [u n : n N($)]) for every $>0. Therefore we obtain that u 0 # S(X) and & 12 (u+u 0 )&=1. From the strict convexity of X it follows u=u 0 ; thus, w-lim u n =u. n Now let x n x and s-lim W A n =A, where A n (n 1) and A are n proximinal convex subsets of X. If y n # P An (x n ) for every n and z # P A (x), from the proof of Lemma 3.1 we know that when x A, then for each
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489
fixed $>0 there is an integer N($) such that co([(x&z)d(x, A)] _ [(x& y n )(d(x, A n )+2 &x&x n &) : n N($)]) & (1&$) U(X )=<. Consequently, w-lim y n =z # P A (x). If x # A, by (4) in Lemma 3.1 we have n y n x # P A (x). Observe that if X is strictly convex and A is a proximinal convex subset, then A is a Chebyshev set. Therefore w-lim P An (x n )=P A (x). n Recall that a reflexive Banach space has the property (C-III) and lim M A n =A implies s-lim W A n =A. Thus we have the following corollaries n n of Theorem 3.2. Corollary 3.4 [12]. Let X be a reflexive and strictly convex Banach space, A n (n 1) and let A be closed convex subsets of X with lim M A n =A. n Then w-lim P An (x)=P A (x) for each x in X. n
Corollary 3.5 [13]. Let X be a Banach space with the property (C-III) and A be a proximinal convex subset in X. Then the mapping x P A (x) is weakly upper semicontinuous. Remark 3.1. Observe that a Banach space X has the property (C-I) if and only if X has the property (C-II) and X is strictly convex. By Theorem 3.1 it is clear that if X has the property (C-I), A n (n 1) and A are proximinal convex sets with s-lim W A n =A, then for each x in X, n lim &P An (x n )&P A (x)&=0 whenever x n x. n
Remark 3.2. Theorem 3.1 is true for (LNUC) or (CL-kR) and Theorem 3.2 is true for (WCL-kR). Remark 3.3. In [11], F. Sullivan introduced the locally k uniformly rotund (Lk-UR) spaces. He proved that if M is a Chebyshev subspace of the (L2-UR) space, then the mapping x P M (x) is norm continuous. In 1985, Yu Xintai [16] extended Sullivan's theorem [11] to (Lk-UR) spaces. From Theorem 1 of [6] it is known that every (Lk-UR) space is (CL-kR). According to Remark 3.2 we get that Theorem 3.1 is true replacing the property (C-II) by (Lk-UR). Consequently, our results generalize the continuity theorems of [11, 16]. REFERENCES 1. J. M. Borwein and S. Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), 843851. 2. B. Brosowski, F. Deutsch, and G. Nurnberger, Parametric approximation, J. Approx. Theory 29 (1980), 261271. 3. J. Diestel, ``Geometry of Banach Spaces-Selected Topics,'' Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, BerlinHeidelbergNew York, 1975. 4. D. Kutzarova and Bor-Luh Lin, Locally k-nearly uniformly convex Banach spaces, Math. Balkanica 8, No. 23 (1994), 203210.
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5. Na Qiyuan, On fully convex and locally fully convex Banach space, Acta Math. Sci. 10, No. 3 (1990), 327343. 6. Nan Chao-Xun and Wang Jian-Hua, On the Lk-UR and L-kR spaces, Math. Proc. Cambridge Philos. Soc. 104 (1988), 521526. 7. B. B. Panda and O. P. Kapoor, A generalization of local uniform convexity of the norm, J. Math. Anal. Appl. 52 (1975), 300308. 8. N. S. Papageorgion and D. A. Kandilakis, Convergence in approximation and nonsmooth analysis, J. Approx. Theory 49 (1987), 4154. 9. J. Rainwater, Local uniform convexity of Day's norm on c 0 (1 ), Proc. Amer. Math. Soc. 22 (1969), 335339. 10. I. Singer, ``Best Approximation in Normed-Linear Spaces by Elements of Linear Subspaces,'' Springer-Verlag, Berlin, 1973. 11. F. Sullivan, A generalization of uniformly rotund Banach spaces, Canad. J. Math. 31 (1979), 628646. 12. M. Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), 301309. 13. Wang Jianhua, Some results on the continuity of metric projections, Math. Appl. 8, No. 1 (1995), 8084. 14. Wang Jianhua and Nan Chao-Xun, On the convergence of =-approximation, in ``Banach Space Theory and Its Application,'' Wuhan Univ. Press, 1996. 15. Wang Jianhua and Wang Musan, Compactly locally fully convex spaces, Kexue Tongbao 36 (1991), 796. 16. Yu Xintai, On LKUR spaces, Chinese Ann. Math. (Ser. B) 6, No. 4 (1985), 465469.
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