sequentially complete in the weak topology
Algebraic properties implying weak sequential completeness
Let \(F\) be an FK-space of real-valued sequences and \(W_F\) be the set of all \(t= (t_k)\in F\) for which \(t= \sum^\infty_{k=1} t_k e^{(k)}\) converges in the weak topology \(\sigma(F, F')\) of \(F\), where, as usual, \(e^{(k)}= (0,\dots,0,1,0,\dots)\) with \(1\) in the \(k\)th place and \(0\) everywhere else.NEWLINENEWLINENEWLINEIt has been established [Bennet-Kalton (1972), Ruckle (1979)] that if \(c_0\subseteq F\) then \(\ell^1\) is sequentially complete in the weak topology \(\sigma(\ell^1, W_F\cap\ell^\infty)\). The sequential completeness of certain sequence spaces under various topologies can be derived from certain algebraic properties of the sequence spaces. It seems therefore natural to ask whether the sequential completeness in the weak topology \(\sigma(\ell^1, W_F\cap \ell^\infty)\) could be guaranteed by some algebraic property of \(W_F\).NEWLINENEWLINENEWLINEIn this paper the author isolates such a property satisfied by \(W_F\cap \ell^\infty\) and gives a generalization of the Bennet-Kalton-Ruckle result.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00019].
scientific article; zbMATH DE number 1619485
scientific article