UDC 513.88:513.83

MATHEMATICS

E. V. OSHMAN

CONTINUITY OF THE METRIC PROJECTION AND SOME GEOMETRIC PROPERTIES OF THE UNIT SPHERE IN A BANACH SPACE

(Presented by Academician A. N. Tikhonov, 1 VII 1968)

Let (E) and (F) be metric spaces: (2^F) is the set of all closed subsets of the space (F). A mapping (\varphi: E \to 2^F) is called a multivalued mapping from (E) into (F). (\varphi) is called upper semicontinuous ((^3)) if the set ({x \in E: \varphi x \subset G}) is open in (E) for every open subset (G \subset F). We shall call the mapping (\varphi) (H)-upper semicontinuous (or upper semicontinuous in the Hausdorff sense) if (x_n \to x) implies
[
\rho(\varphi x_n,\varphi x)=\sup{\rho(y,\varphi x): y \in \varphi x_n}\to 0^*.
]
Both of these definitions coincide with the usual definition of continuity if (\varphi) is single-valued.

Let (M) be a set in (E). Denote by (T_M) the multivalued mapping (E \to 2^M) which assigns to each point (x \in E) the set
[
T_M x={y \in M:\rho(x,y)=\rho(x,M)};
]
(T_M) is called the metric projection of (E) onto (M) ((^2)). Everywhere in what follows: (X) is a (B)-space over the field of real numbers; (S={x \in X:|x|=1}), (S^={f \in X^:|f|=1}); (x_n \rightharpoonup x_0), if the sequence ({x_n}) converges weakly to (x_0); (f_n \rightharpoonup f_0), if (f_n(x)\to f_0(x)) for every (x \in X).

Definition 1. We shall say that (X) satisfies condition (EPS) if from ({x_n}\subset S), (x_0\in S), ({f_n}\subset S^), (f_0\in S^) ((f_0\ne f_n,\ n=1,2,\ldots)), (f_n(x_n)=f_0(x_0)=1), (x_n\rightharpoonup x_0), (f_n\rightharpoonup f_0), and (\rho(x_0,H_0\cap H_n)\to 0), (\rho(x_n,H_0\cap H_n)\to 0), where (H_0={x:f_0(x)=1}), (H_n={x:f_n(x)=1}), it follows that (x_n\to x_0).

Definition 2. We shall say that (X) satisfies condition (CR) (compactly rotund) (respectively (WCR) (weakly compactly rotund)) if for every (f\in S^*) the set ({x:f(x)=1}\cap S) is either empty or compact (respectively, it is either empty or weakly compact).

Definition 3. We shall say that (X) satisfies condition (HR) ((H)-rotund) (respectively (WHR) (weakly (H)-rotund)) if for every (f\in S^) the set ({x:f(x)=1}\cap S) is either empty or satisfies the following condition: from (f(y_n)=1), (\rho(y_n,{x:f(x)=1}\cap S)\to 0) it follows that
[
\rho(y_n,\overline{\operatorname{co}}{y_n}\cap S)\to 0^{*}
]
(respectively, it is either empty, or satisfies the following condition: from (f(y_n)=1), (\rho(y_n,{x:f(x)=1}\cap S)\to 0) and the weak compactness of ({y_n}), it follows that (\rho(y_n,\overline{\operatorname{co}}{y_n}\cap S)\to 0)).

If (A) and (B) are some properties of the space (X), then by ((\mathrm{A})\wedge(\mathrm{B})) we shall denote the property of the space (X) which consists in the fact that (X) possesses both property (A) and property (B).

Theorem 1. In order that, in a reflexive (B)-space (X), the metric projection onto every convex closed set be (H)-semi-

* By definition, we assume that (\rho(y,\Phi)=\infty) for every (y\in F).

** By (\overline{\operatorname{co}}{y_n}) we denote the closed convex hull of the sequence ({y_n}).*

continuous from above, it is necessary and sufficient that (X) satisfy condition ((\mathrm{EPS}) \land (\mathrm{HR})).

A set (M \subset X) is called Chebyshev ((^{1})) if, for every point (x \in X), the set (T_M x) is a singleton.

Corollary 1. In a reflexive (B)-space (X) satisfying condition ((\mathrm{EPS}) \land (\mathrm{HR})), the metric projection onto every convex Chebyshev set is continuous.

Theorem 2. In a reflexive (B)-space (X) the following assertions are equivalent:

(a) the metric projection onto every convex closed set is upper semicontinuous;

(b) (X) satisfies condition ((\mathrm{EPS}) \land (\mathrm{CR}));

(c) (X) satisfies the following condition:

((\mathrm{N})) From ({x_n} \subset S,\ x_0 \in S,\ {f_n} \subset S^,\ f_0 \in S^,\ f_n(x_n)=f_0(x_0)=1,\ x_n \to x_0,\ f_n \to f_0) and (\rho(x_0, H_0 \cap H_n) \to 0,\ \rho(x_n, H_0 \cap H_n) \to 0), where (H_0={x:\ f_0(x)=1},\ H_n={x:\ f_n(x)=1}), it follows that (x_n \to x_0).

Corollary 2. In order that, in a strictly convex and reflexive (B)-space (X), the metric projection onto every convex closed set be continuous, it is necessary and sufficient that (X) satisfy the following condition:

((\mathrm{N_R})) From ({x_n} \subset S,\ x_0 \in S,\ {f_n} \subset S^,\ f_0 \in S^,\ f_n(x_n)=f_0(x_0)=1,\ f_n \to f_0,\ \rho(x_0,H_0\cap H_n)\to 0,\ \rho(x_n,H_0\cap H_n)\to 0), where (H_0(x:\ f_0(x)=1),\ H_n={x:\ f_n(x)=1}), it follows that (x_n \to x_0).

A set (M \subset X) is called boundedly weakly bicompact if its intersection with every closed ball is weakly bicompact.

Theorem 3. Let (X) be a separable (B)-space. Then, in order that in (X) the metric projection onto every boundedly weakly bicompact convex set be (H)-upper semicontinuous, it is necessary and sufficient that (X) satisfy condition ((\mathrm{EPS}) \land (\mathrm{WHR})).

Corollary 3. In a separable (H)-space (X) satisfying condition ((\mathrm{EPS}) \land (\mathrm{WHR})), the metric projection onto every boundedly weakly bicompact convex Chebyshev set is continuous.

Theorem 4. In a separable (B)-space (X) satisfying condition ((\mathrm{WCR})), the following assertions are equivalent:

(a) the metric projection onto every boundedly weakly bicompact convex set is upper semicontinuous;

(b) (X) satisfies condition ((\mathrm{EPS}) \land (\mathrm{CR}));

(c) (X) satisfies condition ((\mathrm{N})).

According to I. Singer ((^{4})), a (B)-space (X) has the Efimov–Stechkin property if it is reflexive and from ({x_n} \subset S,\ x_0 \in S,\ x_n \to x_0) it always follows that (x_n \to x_0). I. Singer ((^{4})) showed that in a (B)-space with the Efimov–Stechkin property the metric projection onto every convex closed set is upper semicontinuous (and hence (H)-upper semicontinuous); in particular, the metric projection onto every convex Chebyshev set is continuous.

Example. Consider in the space (l_2) of real square-summable numerical sequences (x={\xi_i}) the following norm, equivalent to the original one:

[
|x|{S'}=\inf |\lambda|,
]

where

[
S'=\left{x={\xi_i}\in l_2:\ \sum_{i=1}^{\infty}\xi_i^2 \le 1,\ |\xi_i|\le \tfrac12\right}.
]

Denote by (\widetilde l_2) the space (l_2) with norm (|x|_{S'}). Then (\widetilde l_2) is a separable reflexive (B)-space not possessing the Efimov–Stechkin property (see, for example, ((^{4}))). Moreover, in the space (\widetilde l_2) the metric projection onto the closed hyperplane (H={x={\xi_i}:\ \xi_1=\tfrac12}) is not upper semicontinu-

from above. However, it can be shown that $\widetilde{l}_2$ satisfies condition (EPS) $\land$ (HR) and, consequently, by Theorem 1, in it the metric projection onto every convex closed set is $H$-upper semicontinuous; in particular, the metric projection onto every convex Chebyshev set is continuous.

Ural State University
named after A. M. Gorky

Received
31 V 1968

REFERENCES

1. N. V. Efimov, S. B. Stechkin, DAN, 118, No. 1, 17 (1958).
2. V. L. Klee, Math. Ann., 142, 292 (1961).
3. E. Michael, Trans. Am. Math. Soc., 71, 152 (1951).
4. J. Singer, Rev. Roumaine de Math. pure et appl., 9, 167 (1964).