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.
De Bernardi, C. A., Vesely, L., Extension of Continuous Convex Functions from Subspaces II, <<JOURNAL OF CONVEX ANALYSIS>>, 2015; 22 (1): 101-116 [http://hdl.handle.net/10807/113763]
Extension of Continuous Convex Functions from Subspaces II
De Bernardi, Carlo Alberto;
2015
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.