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Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

Peter G. Casazza·Nigel J. Kalton
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Abstract

Let $X$ be a Banach space with an unconditional finite-dimensional Schauder decomposition $(E_n)$. We consider the general problem of characterizing conditions under which one can construct an unconditional basis for $X$ by forming an unconditional basis for each $E_n.$ For example, we show that if $\sup \dim E_n<\infty$ and $X$ has Gordon-Lewis local unconditional structure then $X$ has an unconditional basis of this type. We also give an example of a non-Hilbertian space $X$ with the property that whenever $Y$ is a closed subspace of $X$ with a UFDD $(E_n)$ such that $\sup\dim E_n<\infty$ then $Y$ has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved.

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Peter G. Casazza, Nigel J. Kalton. 1994-08-24. Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces. https://arxiv.org/abs/math/9408207

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