Abstract
From the formal point of view, the goal of this paragraph is to derive Convex-valued selection theorem (1.5) from the 0-dimensional selection theorem (2.4). The value of a continuous singlevalued selection of a given convex-valued mapping will be constructed as the value of an integral (or the barycenter) of a continuous selection of some closed-valued mapping, with respect to a probability measure, defined on some 0-dimensional compactum. This approach is based on the existence of so-called Milyutin mappings. Originally, such kind of maps were applied in the proof of the following (highly nontrivial) fact:
Theorem (3.1). Banach spaces of continuous functions on metrizable uncountable compacta are pairwise isomorphic. In particular, each one of them is isomorphic to the Banach space of continuous functions on the interval.
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© 1998 Springer Science+Business Media Dordrecht
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Repovš, D., Semenov, P.V. (1998). Relations Between Zero-Dimensional and Convex-Valued Selection Theorems. In: Continuous Selections of Multivalued Mappings. Mathematics and Its Applications, vol 455. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1162-3_4
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DOI: https://doi.org/10.1007/978-94-017-1162-3_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5111-0
Online ISBN: 978-94-017-1162-3
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