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

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

Zbl 0295.54047
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