Three mapping theorems. (English) Zbl 0178.25901

The author proves a theorem on the isometric imbedding of a metric space in a Banach space and derives from this theorem a result on continuous selections. The latter is the first theorem of this kind in which the range is nonmetrizable. It goes as follows: if \(X\) is metrizable, if \(M\) is a metrizable subset of a locally convex space \(F\), such that the closed convex hull of every compact subset of \(M\) is compact, and if \(\varphi: X\to 2^N\) is lower-semicontinuous (i.e. \(\varphi(x)\ne\emptyset\) and \(\{x\in X; \varphi(x)\cap V \ne\emptyset\}\) open in \(X\) for every open \(V\subset M)\), and if for some metric on \(M\), every \(\varphi(x)\) is complete, then there exists a continuous \(f: X\to F\) such that \(f(x\in \overline{(\operatorname{conv}\varphi(x))}\) for every \(x\in X\). Moreover, \(f\) may be chosen so that, for every compact \(K\subset X\), \(f(K)\subset \overline{(\operatorname{conv}K')}\) for some compact \(K' \subset \cup \{\varphi(x); x\in K\}\).
From this theorem a result on linear mappings between function spaces is derived, which generalizes an earlier result of the author [Ann. Math. (2) 63, 361–382 (1956; Zbl 0071.15902)].
Reviewer: P. Wuyts


54-XX General topology


Zbl 0071.15902
Full Text: DOI


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