Networks for the weak topology of Banach and Fréchet spaces

Abstract

We start the systematic study of Fréchet 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 Fréchet lcs, then $E$ endowed with its weak topology $\sigma (E,E^{\prime })$ is an $\aleph$-space if and only if $(E,\sigma (E,E^{\prime }))$ is an ${\aleph }_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an $\aleph$-space in the weak topology. We prove that a reflexive Fréchet lcs $E$ in the weak topology $\sigma (E,E^{\prime })$ is an $\aleph$-space if and only if $(E,\sigma (E,E^{\prime }))$ 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.

18 pages

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