First class selectors for weakly upper semi-continuous multi-valued maps in Banach spaces. (English) Zbl 0573.54012

Hansell, R. W.; Jayne, J. E.; Talagrand, Michel

J. Reine Angew. Math. 361, 201-220 (1985).

Let \(X\) be a complete metric space, \(K\) a compact Hausdorff space, \(C_ p(K)\) the space of continuous real-valued functions on \(K\) with the topology of pointwise convergence, and let \(F: X\to C_ p(K)\) be an upper semi-continuous map with nonempty compact values. Although F rarely will have a continuous selector, it is shown that \(F\) will always have a selector which is a pointwise limit of a sequence of continuous functions, both properties holding relative to the Banach space \(C(K)\) under the supremum norm. As a special case, for any Banach space \(E\), if \(f: X\to (E\),weak) is continuous, then \(f: X\to (E\),norm) is Baire class 1. The paper also obtains similar results under weaker assumptions involving the concepts of a Namioka space (in connection with the domain), and the concept of a space being ”fragmented” by a metric (in connection with the range).
{A slight error has been found in theorem 6 of the paper by the first author in Trans. Am. Math. Soc. 194, 195-211 (1974; Zbl 0295.54047) which affect Theorem 1’ of the present paper. All is made well by assuming the metric space \((Y,d)\) is an absolute retract, which keeps intact all previous applications dealing with the important case when \(Y\) is a convex subset of a locally convex metrisable linear space.}

MSC:

54C65	Selections in general topology
46B22	Radon-Nikodým, Kreĭn-Milman and related properties
28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections
54C60	Set-valued maps in general topology
54C35	Function spaces in general topology

Keywords:

upper semi-continuous multi-valued maps; Baire class 1 selectors; Namioka space

Citations:

Zbl 0295.54047