Networks for the weak topology of Banach and Fréchet spaces

0446776 - MÚ 2016 RIV US eng J - Journal Article
Gabriyelyan, S. - Kąkol, Jerzy - Kubiś, Wieslaw - Marciszewski, W.
Networks for the weak topology of Banach and Fréchet spaces.
Journal of Mathematical Analysis and Applications. Roč. 432, č. 2 (2015), s. 1183-1199. ISSN 0022-247X. E-ISSN 1096-0813
R&D Projects: GA ČR(CZ) GA14-07880S
Institutional support: RVO:67985840
Keywords : Fréchet space * space of continuous functions * locally convex space
Subject RIV: BA - General Mathematics
Impact factor: 1.014, year: 2015
http://www.sciencedirect.com/science/article/pii/S0022247X15006836

We start the systematic study of Fréchet spaces which are ...-spaces in the weak topology. A topological space X is an ...-space or an ...-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 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 ...-space if and only if X is countable. We extend Corson's result as follows: If the space ... is a Fréchet lcs, then E endowed with its weak topology sigma(E,E′)sigma(E,E′) is an ...-space if and only if ... is an ...-space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an ...-space in the weak topology.
Permanent Link: http://hdl.handle.net/11104/0248746