Extension of Continuous Convex Functions from Subspaces II

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3 Citations (Scopus)

Abstract

Given Y a subspace of a topological vector space X, and an open convex set 0 is an element of A subset of X, we say that the couple (X, Y) has the CE(A)-property if each continuous convex function on A boolean AND Y admits a continuous convex extension defined on A.Using results from our previous paper, we study for given A the relation between the CE(A)-property and the CE(X)-property. As a corollary we obtain that (X, Y) has the CE(A)-property for each A, provided (X, Y) has the CE(X)-property and Y is "conditionally separable". This applies, for instance, if X is locally convex and conditionally separable. Other results concern either the CE(A)-property for sets A of special forms, or the CE(A)-property for each A where X is a normed space with X/Y separable.In the last section, we point out connections between the CE(X)-property and extendability of certain continuous linear operators. This easily yields a generalization of an extension theorem of Rosenthal, and another result of the same type.
Original languageEnglish
Pages (from-to)101-116
Number of pages16
JournalJournal of Convex Analysis
Volume22
Publication statusPublished - 2015

Keywords

  • Convex function
  • extension
  • normed linear space
  • topological vector space

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