UDC 513.88:513.83

MATHEMATICS

E. B. OSMAN

ON THE CONTINUITY OF THE METRIC PROJECTION ON CERTAIN CLASSES OF SUBSPACES IN A BANACH SPACE

(Presented by Academician A. N. Tikhonov on 27 IV 1970)

Let \(E\) and \(F\) be metric spaces, and let \(K(F)\) be the set of all closed subsets of the space \(F\). A mapping \(\varphi: E \to K(F)\) is called a multivalued mapping from \(E\) into \(F\). \(\varphi\) is called lower semicontinuous \((^3)\) if the set \(\{x \in E: \varphi_x \subset G\}\) is open in \(E\) for every open subset \(G \subset F\), and \(H\)-upper semicontinuous \((^{4,5})\) if for every \(x \in E\) the set \(\varphi_x\) is nonempty and the condition \(x_n \to x\) implies the condition

\[ \widetilde{\rho}(\varphi_{x_n}, \varphi_x)=\sup\{\rho(y,\varphi_x): y \in \varphi_{x_n}\}\to 0. \]

Both definitions coincide with the usual definition of continuity if \(\varphi\) is single-valued. If the mapping \(\varphi\) is upper semicontinuous and \(\varphi_x \ne \varnothing\) for every \(x \in E\), then \(\varphi\) will be \(H\)-upper semicontinuous; moreover, if \(\varphi_x\) is compact for every \(x \in E\), then the converse is also true \((^5)\).

Let \(M\) be a set in \(E\). The multivalued mapping \(T_M\), which assigns to each point \(x \in E\) the set
\[ T_M x=\{y \in M:\rho(x,y)=\rho(x,M)\}, \]
is called the metric projection of \(E\) onto \(M\) \((^2)\). The set \(M\) is called a set of existence \((^1)\) if \(T_M x \ne \varnothing\) for every \(x \in E\), and Chebyshev \((^1)\) if \(T_M x\) is a singleton for every \(x \in E\).

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 \(x_0\); \(\overline{\operatorname{sp}}\{f_n\}\) is the closed linear span of the sequence \(\{f_n\}\subset X^*\); \(\dim \operatorname{sp}\{f_n\}\) is the dimension of the subspace \(\operatorname{sp}\{f_n\}\).

Theorem 1. In a reflexive \(B\)-space \(X\) the following assertions are equivalent:

a) the metric projection onto every closed subspace is lower semicontinuous;

b) \(X\) satisfies the following condition (L): from \(\{x_n\}\subset S\), \(x_0\in S\), \(\{f_n\}_{n=0}^{\infty}\subset S^*\), \(f_n(x_n)=f_0(x_0)=1\), \(x_n \rightharpoonup x_0\), \(\|x_n-x_0\|_{\operatorname{sp}\{f_n\}}\to 0\), where

\[ \|x_n-x_0\|_{\operatorname{sp}\{f_n\}}=\sup\{f(x_n-x_0): f\in \operatorname{sp}\{f_n\}\cap S^*\}, \]

it follows that \(x_n\to x_0\);

c) \(X\) satisfies the following condition (L.1): from \(\{x_n\}\subset S\), \(x_0\in S\), \(\{f_n\}_{n=0}^{\infty}\subset S^*\), \(f_n(x_n)=f_0(x_0)=1\), \(x_n \rightharpoonup x_0\), \(\rho(x_n-x_0,L)\to 0\), where \(L=\bigcap_{n=0}^{\infty}\{x:f_n(x)=0\}\), it follows that \(x_n\to x_0\).

Corollary 1. In a reflexive \(B\)-space \(X\) satisfying condition (L) (or, what is the same, condition (L.1)), the metric projection onto every Chebyshev subspace is continuous.

Theorem 2. In order that in a reflexive \(B\)-space \(X\) the metric projection onto every closed subspace of finite defec-

then (respectively, onto every closed subspace of defect \(\leq k\)) is upper semicontinuous, it is necessary and sufficient that \(X\) satisfy the following condition \((L^\infty)\): from \(\{x_n\}\subset S,\ x_0\in S,\ \{f_n\}_{n=0}^{\infty}\subset S^*,\ f_n(x_n)=f_0(x_0)=1,\ x_n \rightharpoonup x_0,\ \dim \operatorname{sp}\{f_n\}<\infty\) it follows that \(x_n\to x_0\) (respectively, that \(X\) satisfy the following condition \(L^k\): from \(\{x_n\}\subset S,\ x_0\in S,\ \{f_n\}_{n=0}^{\infty}\subset S^*,\ f_n(x_n)=f_0(x_0)=1,\ x_n \rightharpoonup x_0,\ \dim \operatorname{sp}\{f_n\}\leq k\) it follows that \(x_n\to x_0\)).

Corollary 2. In a reflexive \(B\)-space \(X\) satisfying condition \((L^\infty)\) (respectively, satisfying condition \((L^k)\)), the metric projection onto every Chebyshev subspace of finite defect (respectively, onto every Chebyshev subspace of defect \(\leq k\)) is continuous.

Theorem 3. In order that, in a \(B\)-space \(X\), every closed subspace be Chebyshev with a continuous metric projection, it is necessary and sufficient that \(X\) be reflexive and satisfy the following condition \((L_R)\): from \(\{x_n\}\subset S,\ x_0\in S,\ \{f_n\}_{n=0}^{\infty}\subset S^*,\ f_n(x_n)=f_0(x_0)=1,\ \|x_n-x_0\|_{\operatorname{sp}\{f_n\}}\to 0\) it follows that \(x_n\to x_0\) (or, what is the same, condition

\[ (L_R.1):\quad \text{from } \{x_n\}\subset S,\ x_0\in S,\ \{f_n\}_{n=0}^{\infty}\subset S^*,\ f_n(x_n)=f_0(x_0)=1,\ \rho(x_n-x_0,L)\to 0, \]

where

\[ L=\bigcap_{n=0}^{\infty}\{x:\ f_n(x)=0\}, \]

it follows that \(x_n\to x_0\)).

Theorem 4. In order that, in a strictly convex and reflexive \(B\)-space \(X\), the metric projection onto every closed subspace of finite defect (respectively, onto every closed subspace of defect \(\leq k\)) be continuous, it is necessary and sufficient that \(X\) satisfy condition \((L^\infty)\) (respectively, that \(X\) satisfy condition \((L^k)\)).

Theorem 5. In a reflexive \(B\)-space the metric projection onto every closed hyperplane is \(H\)-upper semicontinuous (in a strictly convex space, continuous).

Theorem 6. In a \(B\)-space \(X\) satisfying condition \((L)\), the metric projection onto every reflexive subspace is \(H\)-upper semicontinuous. In particular, the metric projection onto every reflexive Chebyshev subspace is continuous.

A \(B\)-space \(X\) is called compactly rotund \((^4,^5)\) (respectively, weakly compactly rotund \((^4,^5)\)) 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).

Theorem 7. In a weakly compactly rotund \(B\)-space \(X\) satisfying condition \((L)\), the metric projection onto every reflexive subspace is upper semicontinuous.

Theorem 8. In a \(B\)-space \(X\), the metric projection onto every hyperplane that is an existence set is \(H\)-upper semicontinuous.

Theorem 9. In order that, in a \(B\)-space \(X\), the metric projection onto every closed hyperplane be upper semicontinuous, it is necessary and sufficient that \(X\) be compactly rotund.

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

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

where

\[ S'=\left\{x=\{\xi_i\}\in l_2:\ \sum_{i=1}^{\infty}\frac{\xi_i^2}{i^2}\leq 1,\quad \sum_{i=1}^{\infty}\xi_i^2\leq 3\right\}. \]

Denote by \(X\) the space \(l_2\) with the norm \(\|x\|_{S'}\). It is not difficult to see that \(X\) is reflexive, strictly convex, and satisfies condition \((L^\infty)\). Con-

Thus, by virtue of Theorem 4, in the space \(X\) the metric projection onto each closed subspace of finite defect is discontinuous. It can be shown, however, that in \(X\) the metric projection onto some convex closed set is discontinuous.

Ural State University
named after A. M. Gorky
Sverdlovsk

Received
22 IV 1970

REFERENCES

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