Set valued mappings, continuous selections, and metric projections.

*(English)*Zbl 0675.41037Let X,Y be topological spaces. For a set valued mapping \(\phi: X\to 2^ Y\) of X into Y, define \(D(\phi)=\{x\in X:\phi (x)\neq \emptyset \}\), \(G(\phi)=\cup_{x\in X}\{x\}\times \phi(x)\subseteq X\times Y\). If \(\psi: X\to 2^ Y\) and \(\psi(x)\subseteq \phi(x)\) for all \(x\in X\) then \(\psi\) is said to be a submapping of \(\phi\), (symbol. \(\psi\subseteq \phi)\). A continuous selection for \(\phi: X\to 2^ Y\) is a continuous mapping \(s: X\to Y\) such that \(s(x)\in \phi(x)\) for all \(x\in X\). Let \(\phi: X\to 2^ Y\setminus \{\emptyset \}\) be a non-empty set valued mapping. For \(V\subseteq Y\) we put \(L(V):=\{x\in X:\phi(x)\cap V\neq \emptyset\}\), \(U(V):=\{x\in X:\phi(x)\subseteq V\}\) (reviewer’s notation). Then \(\phi\) is said to be lower semi-continuous (respectively upper semi-continuous) if L(V) (respectively U(V)) is an open subset of X whenever V is an open subset of Y. Let \(\phi': X\to 2^ Y\) be defined by \(\phi'(x)=\{y\in \phi(x):x\in \overset\circ L(V)\) whenever \(y\in \overset\circ V,V\subseteq Y\}\) (\(\overset \circ A\) denotes the interior of the set A.) The submapping \(\phi'\) of \(\phi\) is called the derived mapping of \(\phi\). For each ordinal number \(\alpha\) define \(\phi^{(0)}=\phi\), \(\phi^{(\alpha +1)}=(\phi^{(\alpha)})'\), \(\phi^{(\beta)}(x)=\cap_{\alpha <\beta}\phi^{(\alpha)}(x)\) whenever \(\beta\) is a limit ordinal. If card \(\alpha> card X\times Y\) then \(\phi^{(\alpha +1)}=\phi^{(\alpha)}\) and therefore \(\phi^{(\beta)}=\phi^{(\alpha)}\) for all \(\beta\geq \alpha\). The eventual value of the transfinite sequence \(\{\phi^{(\alpha)}:\) \(\alpha\) an ordinal} of derived mappings of \(\phi\) will be denoted by \(\phi^*: X\to 2^ y\) and will be called the stable derived mapping of \(\phi\). Finally, for a real normed linear space X and a subset M of X the metric projection of X onto M is the mapping \(P: X\to 2^ M\), \(P(x)=\{m\in M:\| x-m\| =d(x,M)\}\) where \(d(x,M)=\inf \{\| x- m\|:m\in M\}\) is the distance of x from M. In this very interesting paper the author is concerned with the existence of continuous selections for set valued mappings and particularly those which are metric projections of real normed linear spaces onto finite dimensional subspaces. He characterizes those set valued mappings and investigates the concepts of derived mappings and stable derived mappings for set valued mappings whose values are convex subsets of a finite dimensional real linear space. We state the main theorems of the paper:

Theorem 1. Let X be a finite dimensional real linear space and let M be a subspace of X. If \(P: X\to 2^ M\) is a set valued mapping of X into M then there exists a norm on X such that P is the metric projection of X onto M relative to that norm if and only if the following conditions are satisfied: (i) P: \(X\to 2^ M\) is upper semi-continuous. (ii) P(x) is non-empty, compact and convex for each \(x\in X\). (iii) \(P(\lambda x)=\lambda P(x)\) for all \(x\in X\) and \(\lambda\in R\). (iv) \(P(x+m)=P(x)+m\) for all \(x\in X\) and \(m\in M\).

Theorem 2. Suppose that n is a positive integer, X is a topological space and \(\phi\) is a set valued mapping from X into \(R^ n\) such that \(\phi(x)\) is convex for each \(x\in X\). Then \(\rho^{(n)}| \overset\circ D(\phi^{(n)})\) is lower semi-continuous and, consequently \[ \varphi^* = \begin{cases} \varphi^{(n)} \qquad&\text{if }D(\phi^{(n)}) \text{ is open in }X \\ \varphi^{(n+1)} \qquad&\text{if }D(\phi^{(n)}) \text{ is not open in }X. \end{cases} \] Theorem 3. (a) There exists a real normed linear space X of dimension \(2n+1\) and a subspace M of X, of dimension n, such that for the metric projection P: \(X\to 2^ M\) of X onto M (1) \(D(P^{(n-1)})=X\), (2) \(P^{(n)}=/=P^{(n+1)}\). (b) There exists a normed linear space of dimension 2n and a subspace M of X, of dimension n, such that for the metric projection \(P: X\to 2^ M\) of X onto M (1) \(D(P^{(n)})=X\), (2) \(P^{(n-1)}=/=P^{(n)}\). Theorem 4. If M is any finite dimensional subspace of C(X), of dimension n, and P is the metric projection of C(X) onto M then \(P'\subseteq P_ n\), \(\overset\circ D(P_ 1)=\overset\circ D(P_ n)=\overset\circ D(P')= D(P^*)\), and \(P_ n,P'\) and \(P^*\) coincide on \(D(P^*)\).

Theorem 1. Let X be a finite dimensional real linear space and let M be a subspace of X. If \(P: X\to 2^ M\) is a set valued mapping of X into M then there exists a norm on X such that P is the metric projection of X onto M relative to that norm if and only if the following conditions are satisfied: (i) P: \(X\to 2^ M\) is upper semi-continuous. (ii) P(x) is non-empty, compact and convex for each \(x\in X\). (iii) \(P(\lambda x)=\lambda P(x)\) for all \(x\in X\) and \(\lambda\in R\). (iv) \(P(x+m)=P(x)+m\) for all \(x\in X\) and \(m\in M\).

Theorem 2. Suppose that n is a positive integer, X is a topological space and \(\phi\) is a set valued mapping from X into \(R^ n\) such that \(\phi(x)\) is convex for each \(x\in X\). Then \(\rho^{(n)}| \overset\circ D(\phi^{(n)})\) is lower semi-continuous and, consequently \[ \varphi^* = \begin{cases} \varphi^{(n)} \qquad&\text{if }D(\phi^{(n)}) \text{ is open in }X \\ \varphi^{(n+1)} \qquad&\text{if }D(\phi^{(n)}) \text{ is not open in }X. \end{cases} \] Theorem 3. (a) There exists a real normed linear space X of dimension \(2n+1\) and a subspace M of X, of dimension n, such that for the metric projection P: \(X\to 2^ M\) of X onto M (1) \(D(P^{(n-1)})=X\), (2) \(P^{(n)}=/=P^{(n+1)}\). (b) There exists a normed linear space of dimension 2n and a subspace M of X, of dimension n, such that for the metric projection \(P: X\to 2^ M\) of X onto M (1) \(D(P^{(n)})=X\), (2) \(P^{(n-1)}=/=P^{(n)}\). Theorem 4. If M is any finite dimensional subspace of C(X), of dimension n, and P is the metric projection of C(X) onto M then \(P'\subseteq P_ n\), \(\overset\circ D(P_ 1)=\overset\circ D(P_ n)=\overset\circ D(P')= D(P^*)\), and \(P_ n,P'\) and \(P^*\) coincide on \(D(P^*)\).

Reviewer: C.G.Lascarides

##### MSC:

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |

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\textit{A. L. Brown}, J. Approx. Theory 57, No. 1, 48--68 (1989; Zbl 0675.41037)

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