Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender , which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us to characterize reflexive Banach spaces as the only normed spaces Y such that for every closed subset A of a GO-space X there is a ℂ-extender for the family ℂ of closed convex subsets of Y. Also we obtain a characterization of Polish spaces and of weakly sequentially complete Banach lattices in terms of extenders.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-2-2,
author = {Iryna Banakh and Taras Banakh and Kaori Yamazaki},
title = {Extenders for vector-valued functions},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {123-150},
zbl = {1176.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-2-2}
}
Iryna Banakh; Taras Banakh; Kaori Yamazaki. Extenders for vector-valued functions. Studia Mathematica, Tome 192 (2009) pp. 123-150. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-2-2/