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arXiv · 1907.11863

Schauder Bases Having Many Good Block Basic Sequences

Cory A. Krause
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Abstract

In the study of asymptotic geometry in Banach spaces, a basic sequence which gives rise to a spreading model has been called a good sequence. It is well known that every normalized basic sequence in a Banach space has a subsequence which is good. We investigate the assumption that every normalized block tree relative to a basis has a branch which is good. This combinatorial property turns out to be very strong and is equivalent to the space being $1$-asymptotic $\ell_p$ for some $1\leq p\leq\infty$. We also investigate the even stronger assumption that every block basic sequence of a basis is good. Finally, using the Hindman-Milliken-Taylor theorem, we prove a stabilization theorem which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good, and we present an application of this stabilization.

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Cory A. Krause. 2019-07-27. Schauder Bases Having Many Good Block Basic Sequences. https://arxiv.org/abs/1907.11863

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