Alejandra Carolina Cáceres Rigo: An $\mathcal{A}$-tight - $\mathcal{A}$-minimal dichotomy theorem for Banach spaces.

Data: sexta-feira, 13 de novembro de 2020, às 9h30.

Alejandra Carolina Cáceres Rigo (IME-USP)

Título: An A-tight - A-minimal dichotomy theorem for Banach spaces.

Resumo:  As part of the program of classification of Banach spaces up to subspaces initiated by W. T. Gowers, V. Ferenczi and Ch. Rosendal proved a third dichotomy which states that every Banach space has a subspace that is either minimal or tight. In this work we define the notions of $\mathcal{A}$-minimality and $\mathcal{A}$-tightness over Banach spaces with Schauder basis, considering $\mathcal{A}$ an admissible property over sequences of vectors in a Banach space. We generalize the techniques used in the proof of the third dichotomy to show that given an admissible property $\mathcal{A}$, every Banach space contains a separable Banach space that is either $\mathcal{A}$-minimal or $\mathcal{A}$-tight. We prove that the third dichotomy and other dichotomies given by V. Ferenczi and Ch. Rosendal are obtained as corollaries.

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