Tight-minimal dichotomies in Banach spaces
Tom 281 / 2025
Streszczenie
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 $(\mathcal D_E, \mathcal A_{E})$ contains an infinite-dimensional subspace with a basis which is either $\mathcal A_{E}$-tight or $\mathcal A_{E}$-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 $(e_n)_n$ has a subsequence which 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.
