Networks for the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kakol, W. Kubiś, W. Marciszewski

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We start the systematic study of Fréchet spaces which are ℵ-spaces in the weak topology. A topological space X is an ℵ0-space or an ℵ-space if X has a countable k-network or a σ-locally finite k-network, respectively. We are motivated by the following result of Corson (1966): If the space Cc(X) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology is a Banach space, then Cc(X) endowed with the weak topology is an ℵ0-space if and only if X is countable. We extend Corson's result as follows: If the space E:=Cc(X) is a Fréchet lcs, then E endowed with its weak topology σ(E, E') is an ℵ-space if and only if (E, σ(E, E')) is an ℵ0-space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an ℵ-space in the weak topology. We prove that a reflexive Fréchet lcs E in the weak topology σ(E, E') is an ℵ-space if and only if (E, σ(E, E')) is an ℵ0-space if and only if E is separable. We show however that the nonseparable Banach space ℓ1(R) with the weak topology is an ℵ-space.

Original languageEnglish
Pages (from-to)1183-1199
Number of pages17
JournalJournal of Mathematical Analysis and Applications
Volume432
Issue number2
DOIs
StatePublished - 15 Dec 2015

Keywords

  • Fréchet space
  • Space of continuous functions
  • Weakly ℵ locally convex space
  • ℵ-space
  • ℵ-space

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