Hereditary invertible linear surjections and splitting problems for selections

Details Hereditary invertible linear surjections and splitting problems for selections Hereditary invertible linear surjections and splitting problems for selections

2008-03-29

arxiv.org/abs/0803.4254v1

Topology Appl. 156:7 (2009), 1192-1198.

Mathematics

General Topology

Scientific paper

10.1016/j.topol.2008.12.014

Let $A+B$ be the pointwise (Minkowski) sum of two convex subsets $A$ and $B$ of a Banach space. Is it true that every continuous mapping $h:X \to A+B$ splits into a sum $h=f+g$ of continuous mappings $f:X \to A$ and $g:X \to B$? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.

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