Hereditary invertible linear surjections and splitting problems for selections |
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Authors: | Du&scaron an Repov&scaron ,Pavel V. Semenov |
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Affiliation: | a Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, PO Box 2964, Ljubljana, Slovenia 1001 b Department of Mathematics, Moscow City Pedagogical University, 2-nd Selskokhozyastvennyi pr. 4, Moscow, Russia 129226 |
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Abstract: | 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→A+B splits into a sum h=f+g of continuous mappings f:X→A and g:X→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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Keywords: | primary, 54C60, 54C65, 41A65 secondary, 54C55, 54C20 |
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