On shrinking and boundedly complete Schauder frames of Banach spaces

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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.

Keywords

Associated space
Duality theory
Schauder frame

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This work is supported by funds from the Linear Analysis Workshop at Texas A&M University in 2008, the China Scholarship Council (CSC), the National Natural Science Foundation of China (No. 10571090), and the Doctoral Programme Foundation of Institution of Higher Education (No. 20060055010).