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arXiv · 2503.06880

Characterizations of reflexive Banach spaces

Tianyi Zhou
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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.

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Tianyi Zhou. 2025-03-10. Characterizations of reflexive Banach spaces. https://arxiv.org/abs/2503.06880

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