Besselian Schauder Frames and the Structure of Banach Spaces

TL;DR

This paper explores Besselian Schauder frames (BSF) in Banach spaces, extending classical basis theory, and demonstrates their limitations and constructions, including for spaces lacking Schauder bases.

Contribution

It introduces Besselian Schauder frames as a generalization of Schauder bases, extends classical results to this framework, and provides explicit constructions for Banach spaces.

Findings

Every unconditional Schauder frame is BSF, but not vice versa.

Many classical Banach spaces do not admit BSF or USF.

Constructed Schauder frames for spaces with finite dimensional decompositions.

Abstract

Schauder bases are fundamental tools for analyzing the structure of Banach spaces. In this work, we show that Besselian Schauder frames (BSF) play a similar role in certain contexts. We first prove that every unconditional Schauder frame (USF) is BSF, but the reverse implication is false. Specifically, we extend several well-known results of Karlin and James to Banach spaces with BSF, particularly to those with USF. We prove that many classical Banach spaces do not admit BSF, and in particular, do not admit USF. Before establishing these results, for every Banach space $E$ with a finite dimensional decomposition, we provide an explicit method to construct a Schauder frame for $E$. In particular, Szarek's Banach space has a Schauder frame, which famously lacks a Schauder basis. This finding provides strong motivation for extending classical Schauder basis theory to the framework of Schauder frames.