On Shrinking and Boundedly Complete Schauder Frames of Banach spaces

TL;DR

This paper extends classical duality theorems and characterizations of Banach space properties from bases to Schauder frames, introducing new concepts like minimal and maximal spaces, and generalizing key results by James.

Contribution

It generalizes duality theorems and reflexivity characterizations from bases to Schauder frames in Banach spaces, broadening the theoretical framework.

Findings

Extended duality theorems to Schauder frames

Generalized James' results on shrinking and boundedly complete frames

Characterized reflexivity using unconditional frames

Abstract

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames.