Networks for the weak topology of Banach and Fr\'echet spaces

TL;DR

This paper systematically studies the weak topology of Fréchet and Banach spaces, characterizing when these spaces are $\

Contribution

It extends Corson's classical result, providing necessary and sufficient conditions for Fréchet spaces to be $\\aleph$-spaces in the weak topology, especially linking separability and $\\aleph$-space properties.

Findings

A Fréchet lcs with weak topology is an \\aleph-space iff the underlying space is countable.

A reflexive Fréchet lcs in the weak topology is an \\aleph-space iff it is separable.

The nonseparable Banach space \\ell_{1}(\\mathbb{R}) with weak topology is an \\aleph-space.

Abstract

We start the systematic study of Fr\'{e}chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $\sigma$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows: If the space $E:=C_{c}(X)$ is a Fr\'echet lcs, then $E$ endowed with its weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fr\'echet lcs to be an $\aleph$-space in the weak topology. We prove that a reflexive Fr\'echet lcs $E$ in the weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $E$ is separable. We show however that the nonseparable Banach space $\ell_{1}(\mathbb{R})$ with the weak topology is an $\aleph$-space.