Networks for the weak topology of Banach and Frechet spaces

We start the systematic study of Frechet spaces which are N-spaces in the weak topology. A topological space X is an N-0-space or an N-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 N-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 Frechet les, then E endowed with its weak topology sigma(E, E') is an N-space if and only if (E, sigma(E, E')) is an N-0-space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Frechet lcs to be an N-space in the weak topology. We prove that a reflexive Frechet lcs E in the weak topology sigma(E, E') is an N-space if and only if (E, sigma(E, E')) is an N-0-space if and only if E is separable. We show however that the nonseparable Banach space l(1)(R) with the weak topology is an N-space. (C) 2015 Elsevier Inc. All rights reserved.