Some nonlinear characterizations of reflexive Banach spaces

TL;DR

This paper establishes that various formulations related to the attainment of infima in reflexive Banach spaces are equivalent to the space's reflexivity, and characterizes finite dimensionality via minimization properties of functions.

Contribution

It proves the equivalence of multiple formulations of infima attainment with reflexivity and characterizes finite dimensional spaces through a minimization property.

Findings

All formulations of infima attainment are equivalent to reflexivity.

A normed space is finite dimensional iff every continuous function on bounded closed sets attains its minimum.

Provides a new characterization of finite dimensionality based on minimization properties.

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 obtain its minimum, namely the converse of the classical Weierstrass theorem also holds true.