2023/12/01 by Alejandra C. Cáceres-Rigo, Cáceres-Rigo, Alejandra C., Valentin Ferenczi +1
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2312.00721
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.