Vector Valued Closed Subspaces and Characterizations of Normed Spaces Through (j'-summability

Abstract

Aizpuru and Nicasio-Llach [1] introduced the spaces of vector valued sequences defined by statistical convergence and beside of some new characterizations like completeness, reflexivity and Shur properties of normed spaces, they also obtained a new version of Antosik-Swartz basic matrix theorem. Aizpuru et al. [21 and Kama [131 studied these properties in terms of vector valued ahnost convergence and I-statistical convergence, respectively. Recently, the authors gave some similar results on normed spaces by using a generalization of vector valued almost convergence, [17]. ill the present paper, we essentially deal with invariant means (a-summability) to have some new vector valued closed subspaces of loo(X) and bs(X), and to get some new characterizations of completeness, reflexivity, Schur and Grothendieck properties of nonned spaces. We also give a new characterization of finite dimensionality of normed spaces. © 2024, Allahabad Mathematical Society. All rights reserved.