Symmetric sequence spaces, bases, and applications
The purpose of this paper is three-fold, namely (i) to prove results on symmetric sequence spaces which are not normal, (ii) relate the symmetric character of a Schauder base (S.b.) in a locally convex space (l.c. TVS) with the symmetry of the associated sequence space (a.s.s.); and (iii) look for the applications of these results. For instance, we prove Theorem: If \(\lambda\) is a symmetric sequence space such that \(\phi \subset \lambda \subset c_ 0\), then \(\lambda^ x=\lambda^{\beta}.\) Theorem: An S.b. for a barrelled space is symmetric if and only if the a.s.s. is symmetric. Theorem: Let \(\{x_ n;f_ n\}\) be an s-symmetric s.b. for an l.c. TVS (X,T) such that \(f_ n(x_ 0)\not\in \ell^{\infty}\) for some \(x_ 0\) in x. Then (X,T) is metrizable and if in addition, (X,T) is also sequentially complete, then (X,T) is a Fréchet space and \(\{x_ n\}\) is bounded in (X,T). Besides, there are many other results related to (i), (ii), (iii); and counter-examples in support of assumption of various hypotheses.
symmetric sequence spaces which are not normal
scientific article; zbMATH DE number 3981803
scientific article