Networks for the weak topology of Banach and Fréchet spaces

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 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 ℵ₀-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 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.