Kurt Gödel Research Center

2016 seminar talk: Selected topics for the weak topology of Banach spaces

Talk held by Jerzy Kąkol (Adam Mickiewicz University Poznań, Poland) at the KGRC seminar on 2016-04-21.

Abstract

Corson (1961) started a systematic study of certain topological properties of the weak topology $w$ of Banach spaces $E$. This line of research provided more general classes such as reflexive Banach spaces, Weakly Compactly Generated Banach spaces and the class of weakly $K$-analytic and weakly $K$-countably determined Banach spaces. On the other hand, various topological properties generalizing metrizability have been studied intensively by topologists and analysts. Let us mention, for example, the first countability, Frechet-Urysohn property, sequentiality, $k$-space property, and countable tightness. Each property (apart the countable tightness) forces a Banach space $E$ to be finite-dimensional, whenever $E$ with the weak topology $w$ is assumed to be a space of the above type. This is a simple consequence of a theorem of Schluchtermann and Wheeler that an infinite-dimensional Banach space is never a $k$-space in the weak topology. These results show also that the question when a Banach space endowed with the weak topology is homeomorphic to a certain fixed model space from the infinite-dimensional topology is very restrictive and motivated specialists to detect the above properties only for some natural classes of subsets of $E$, e.g., balls or bounded subsets of $E$. We collect some classical and recent results of this type, and characterize those Banach spaces $E$ whose unit ball $B_w$ is $k_\mathbb{R}$-space or even has the Ascoli property. Some basic concepts from probability theory and measure theoretic properties of the space $\ell_1$ will be used.