Quantifying shrinking and boundedly complete bases
Dongyang Chen, Tomasz Kania, Yingbin Ruan
TL;DR
This paper introduces new quantitative measures for bases in Banach spaces, providing refined versions of James' classical characterizations of reflexivity, especially for spaces with unconditional bases.
Contribution
It develops novel quantitative metrics for bases, extending James' characterizations of reflexivity to a more precise, measurable framework.
Findings
New quantities measuring how far a basis is from being shrinking or boundedly complete.
Quantitative versions of James' characterizations of reflexivity.
Extensions to Banach spaces with unconditional bases.
Abstract
We investigate possible quantifications of R. C. James' classical work on bases and reflexivity of Banach spaces. By introducing new quantities measuring how far a basic sequence is from being shrinking and/or boundedly complete, we prove quantitative versions of James' famous characterisations of reflexivity in terms of bases. Furthermore, we establish quantitative versions of James' characterisations of reflexivity of Banach spaces with unconditional bases.
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Taxonomy
TopicsAdvanced Banach Space Theory · Approximation Theory and Sequence Spaces · Optimization and Variational Analysis