Abstract of a paper by S. Gabriyelyan, J. Kcakol, W. Kubis, and W. Marciszewski
This is an announcement for the paper "Networks for the weak topology of Banach and Frechet spaces" by S. Gabriyelyan, J. Kcakol, W. Kubis, and W. Marciszewski. 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. Archive classification: math.FA Mathematics Subject Classification: Primary 46A03, 54H11, Secondary 22A05, 54C35 Remarks: 18 pages Submitted from: kubis@math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.1748 or http://arXiv.org/abs/1412.1748
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alspach@math.okstate.edu