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Sem. Topologia i T. Mnogości


Overcomplete sets

Prelegent: Piotr Koszmider

2022-12-14 16:15

The density of a topological vector space (tvs) X is the minimal cardinality of a dense subset of X.
A subset of a tvs is called linearly dense if the set of all  linear combinations of its elements forms a dense subset.
A subset Y of a tvs X is called overcomplete if it has cardinality equal to the density of X and every subset  of Y of this cardinality is linearly dense in X.

By a classical result of Klee every separable Banach space admits an overcomplete set. We will address the issue of the existence of overcomplete sets in nonseparable Banach spaces, showing first ZFC examples of such spaces admitting overcomplete sets, discussing independence and consistency results and some negative results. Many basic problems remain open, for example, if it is consistent that there is a Banach space of density bigger than omega_1 admitting an overcomplete set, or if the space of continuous functions on the ladder system space admits an overcomplete set in ZFC.

The talk is based on the paper: P. Koszmider, On the existence of overcomplete sets in some classical nonseparable Banach spaces, J. Funct. Anal. 281 (2021), no. 9, Paper No. 109172, 33 pp. We will not assume any knowledge of analysis or topology beyond  basics from the undergraduate level like the the identity theorem for analytic functions, Hahn-Banach theorem or the Baire category theorem.