Some nonlinear characterizations of reflexive Banach spaces

Yan Tang, Shiqing Zhang, Tiexin Guo

Published 2021-10-11, updated 2022-01-21Version 4

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. A similar characterization of a finite--dimensional normed space is also given in passing for an interesting contrast. Besides, several new basic results of lower semicontinuous convex functions are found in the course of our work and play a crucial role in the proofs of the main results.