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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

MSC:

54-XX General topology

Citations:

Zbl 0071.15902
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References:

[1] N. Bourbaki, Eléments de mathématique. XV. Première partie: Les structures fondamentales de l’analyse. Livre V: Espaces vectoriels topologiques. Chapitre I: Espaces vectoriels topologiques sur un corps valué. Chapitre II: Ensembles convexes et espaces localement convexes, Actualités Sci. Ind., no. 1189, Herman & Cie, Paris, 1953 (French). · Zbl 0050.10703
[2] -, Espaces vectoriels topologiques, Chapters III and IV, Hermann, Paris, 1955. · Zbl 0068.09001
[3] Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152 – 182. · Zbl 0043.37902
[4] Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361 – 382. · Zbl 0071.15902
[5] E. Michael, A theorem on semi-continuous set-valued functions, Duke Math. J 26 (1959), 647 – 651. · Zbl 0151.30805
[6] E. Michael, A short proof of the Arens-Eells embedding theorem, Proc. Amer. Math. Soc. 15 (1964), 415 – 416. · Zbl 0178.25903
[7] E. Michael, A linear mappping between function spaces, Proc. Amer. Math. Soc. 15 (1964), 407 – 409. · Zbl 0133.07204
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