On the global shape of convex functions on locally convex spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:488

Article

Zalinescu, C.¹ 

[1] Romanian Acad, Iasi Branch, Octav Mayer Inst Math, Bd Carol I,8, Iasi 700505, Romania

关键词: Convex function;    Coercivity;    Recession function;    Quasi relative interior;    Locally convex space;   

DOI  :  10.1016/j.jmaa.2020.124109

来源: Elsevier

【 摘 要 】

In the recent paper [1] D. Azagra studies the global shape of continuous convex functions defined on a Banach space X. More precisely, when X is separable, it is shown that for every continuous convex function f : X -> R there exist a unique closed linear subspace Y of X, a convex function h: X/Y -> R with the property that lim(t ->infinity) h(u + tv) = infinity for all u,v is an element of X/Y, v not equal 0, and x* is an element of X* such that f = h circle pi + x*, where pi : X -> X/Y is the natural projection. Our aim is to characterize those proper lower semicontinuous convex functions defined on a locally convex space which have the above representation. In particular, we show that the continuity of the function f and the completeness of X can be removed from the hypothesis of Azagra's theorem. For achieving our goal we study general sublinear functions as well as recession functions associated to convex ones. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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