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Open AccessArticle

Pre-Schauder Bases in Topological Vector Spaces

Department of Mathematics, University of Cadiz, 11519 Puerto Real, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2019, 11(8), 1026; https://doi.org/10.3390/sym11081026
Received: 17 June 2019 / Revised: 18 July 2019 / Accepted: 2 August 2019 / Published: 9 August 2019
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
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Abstract

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n N in X such that for every x X there exists a unique sequence of scalars ( λ n ) n N satisfying that x = n = 1 λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized. View Full-Text
Keywords: Schauder basis; topological vector space; monotone basis; Hausdorff topology Schauder basis; topological vector space; monotone basis; Hausdorff topology
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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García-Pacheco, F.J.; Pérez-Fernández, F.J. Pre-Schauder Bases in Topological Vector Spaces. Symmetry 2019, 11, 1026.

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