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arXiv · 2312.00721

Tight-minimal dichotomies in Banach spaces

Alejandra C. Cáceres-Rigo·Valentin Ferenczi
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Abstract

We extend the methods used by V. Ferenczi and Ch. Rosendal to obtain the `third dichotomy' in the program of classification of Banach spaces up to subspaces, in order to prove that a Banach space E with an admissible system of blocks with admissible set A, contains an infinite dimensional subspace with a basis which is either A-tight or A-minimal. In this setting we obtain, in particular, dichotomies regarding subsequences of a basis, and as a corollary, we show that every normalized basic sequence has a subsequence which either satisfies a tightness property or is spreading. Other dichotomies between notions of minimality and tightness are demonstrated, and the Ferenczi-Godefroy interpretation of tightness in terms of Baire category is extended to this new context.

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Alejandra C. Cáceres-Rigo, Valentin Ferenczi. 2023-12-01. Tight-minimal dichotomies in Banach spaces. https://arxiv.org/abs/2312.00721

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