Characterizations of reflexive Banach spaces
Tianyi Zhou
TL;DR
This survey synthesizes renowned equivalences linking reflexivity to convergence of arithmetic means, weak compactness, affine geometry, Schauder-basis properties, bi-orthogonal systems, and dual-space behavior. It presents core dual-space criteria, affine-set and weak-compactness characterizations, and basis-based obstructions (notably subspaces isomorphic to $\ell_1$ or $c_0$) as decisive indicators of non-reflexivity. It also delineates super-properties such as uniform non-square and uniform convexifiability, and their intricate connections to the von Neumann–Jordan constant and super-reflexivity. The work emphasizes that reflexivity sits at the intersection of geometric, combinatorial, and functional-analytic structures, with wide implications for analyzing subspaces, duals, and norm-equivalent reformulations across 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 Cesàro 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.