Seminar: Working group in applications of set theory, IMPAN

Thursday, 20.12.2018, 10:15, room 105, IMPAN

Speaker: Damian Sobota (Kurt Godel RC, Vienna)

Title: “The Josefson–Nissenzweig theorem for Cp(X)-spaces ”

Abstact: “The famous Josefson–Nissenzweig theorem asserts that for every infinite-dimensional Banach space X there exists a sequence (x_n*) in the dual space X* which is weak* convergent to 0 and each x_n* has norm 1. Despite the apparent simplicity of the theorem no constructive proof — even in the case of Banach spaces of continuous functions on compact spaces — has been known.

Recently, Banakh, Śliwa and Kąkol in their studies of separable quotients of topological vector spaces of the form Cp(X), i.e. spaces of continuous functions on Tychonoff spaces endowed with the pointwise convergence topology, have obtained several results characterizing those Cp(X)-spaces for which the Josefson–Nissenzweig theorem holds.

During my talk I will present some introductory facts concerning the theorem for Cp(X)-spaces, show that the existence of “Josefson–Nissenzweig” sequences for Cp(K)-spaces, where K is compact Hausdorff, is strongly related to a variant of the Grothendieck property of Banach spaces, as well as prove that every compact space obtained as a limit of an inverse system consisting only of minimal extensions admits such sequences (and the proof is constructive). This is a joint work with Lyubomyr Zdomskyy”.

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