RT journal article
T1 Pre-Schauder Bases in Topological Vector Spaces
A1 García Pacheco, Francisco Javier
A1 Pérez Fernández, Francisco Javier
A2 Matemáticas
K1 Schauder basis
K1 topological vector space
K1 monotone basis
K1 Hausdorff topology
AB A Schauder basis in a real or complex Banach space X is a sequence (en)n is an element of N in X such that for every x is an element of X there exists a unique sequence of scalars (lambda n)n is an element of N satisfying that x= n-ary sumation n=1 infinity lambda nen. 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.
PB MDPI
SN 2073-8994
YR 2019
FD 2019-08
LK http://hdl.handle.net/10498/21741
UL http://hdl.handle.net/10498/21741
LA eng
DS Repositorio Institucional de la Universidad de Cádiz
RD 13-nov-2026