# Tomasz Kochanek: Bases of Banach spaces with respect to filters

Seminar: Working group in applications of set theory, IMPAN

Thursday, 22.11. 2018, 10:15, room 105, IMPAN

Speaker: Tomasz Kochanek (IM PAN / MIM UW)

Title: “Bases of Banach spaces with respect to filters”

Abstact: “In 2011, Vladimir Kadets proposed the following problem: Given a filter F of subsets of natural numbers and a Banach space X, we say that a sequence (e_n) in X forms an F-basis, provided that every x in X has a unique representation as a series of linear combinations of e_n’s, where the convergence is understood in the norm topology and with respect to F. Thus, for F being the filter of cofinite sets we obtain the classical notion of Schauder basis for which it is well-known that all the coordinate functionals are automatically continuous. The question is whether, they must be continuous for a general filter F. I shall present a positive answer to this questions in the case where the character of F is smaller than the pseudointersection number (published in Studia Math. 2012). Unfortunately, the answer is still not known in the important case where F is the statistical filter consisting of all sets of asymptotic density 1. We will also discuss some other related open problems concerning bases with brackets and with individual brackets”.

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