Release time:2024-10-25
Impact Factor:1.3
DOI number:10.1016/j.jmaa.2022.126736
Journal:J. Math. Anal. Appl.
Key Words:Reflexive Banach spaces inf–convolution James theorem Lower semicontinuous convex coercive function Sequentially weakly lower semicontinuous coercive function Attainment of infima
Abstract:It is well known that in the calculus of variations and in optimization there exist many formulations of the fundamental propositions on the attainment of the infima of sequentially weakly lower semicontinuous coercive functions on reflexive Banach spaces. By either some constructive skills or the regularization skill by inf–convolutions we show in this paper that all these formulations together with their important variants are equivalent to each other and equivalent to the reflexivity of the underlying space. Motivated by this research, we also give a characterization for a normed space to be finite dimensional: a normed space is finite dimensional iff every continuous real–valued function defined on each bounded closed subset of this space can attain its minimum, namely the converse of the classical Weierstrass theorem also holds true.
Co-author:Shiqing Zhang, Tiexin Guo
First Author:Yan Tang
Indexed by:Journal paper
Document Code:126736
Discipline:Natural Science
First-Level Discipline:Mathematics
Document Type:J
Volume:519
ISSN No.:0022-247X
Translation or Not:no
Date of Publication:2022-09-30
Included Journals:SCI
Links to published journals:https://www.sciencedirect.com/science/article/pii/S0022247X22007508?via%3Dihub
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