Characterizations of reflexive Banach spaces

Abstract

In this paper we survey known results of characterizations of reflexive Banach spaces, which are based on convergence of usual and generalized arithmetic mean (or Ces\`aro sum), weakly compact subsets, affine sets in a Banach space or its dual and an unbounded bi-orthogonal system generalized from the one in a finite-dimensional Banach space. We also include results that describe precisely when a subspace is linearly isomorphic to $\ell^1$ or $c_0$ in a Banach space that has a Schauder basis, which can imply non-reflexivity of a Banach space in general and is proven to be equivalent to non-reflexivity when the given Schauder basis is unconditional. After reflexivity, we will also study other geometric properties that are strictly stronger, implications among them and their characterizations.