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 ℵ0ℵ0-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)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)Cc(X) endowed with the weak topology is an ℵ0ℵ0-space if and only if X   is countable. We extend Corson's result as follows: If the space E:=Cc(X)E:=Cc(X) is a Fréchet lcs, then E   endowed with its weak topology σ(E,E′)σ(E,E′) is an ℵ-space if and only if (E,σ(E,E′))(E,σ(E,E′)) is an ℵ0ℵ0-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′)σ(E,E′) is an ℵ-space if and only if (E,σ(E,E′))(E,σ(E,E′)) is an ℵ0ℵ0-space if and only if E   is separable. We show however that the nonseparable Banach space ℓ1(R)ℓ1(R) with the weak topology is an ℵ-space.