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Pre-Schauder Bases in Topological Vector Spaces

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Pre-Schauder Bases in Topological Vector Spaces S S symmetry Article Pre-Schauder Bases in Topological Vector Spaces Francisco Javier García-Pacheco *,† and Francisco Javier Pérez-Fernández † Department of Mathematics, University of Cadiz, 11519 Puerto Real, Spain * Correspondence: [email protected] † These authors contributed equally to this work. Received: 17 June 2019; Accepted: 2 August 2019; Published: 9 August 2019 Abstract: A Schauder basis in a real or complex Banach space X is a sequence (en)n2N in X such ¥ that for every x 2 X there exists a unique sequence of scalars (ln)n2N satisfying that x = ∑n=1 lnen. 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. Keywords: Schauder basis; topological vector space; monotone basis; Hausdorff topology 1. Introduction Schauder bases were introduced for the first time in [1] in the setting of real or complex Banach spaces. However, they can be transported to a more general scope: the category of topological vector spaces over an absolutely valued division ring. We refer the reader to [2–5] for a more general perspective on this category. A topological vector space X over an absolutely valued division ring K is said to have a pre-Schauder basis (en)n2N ⊂ X provided that for every x 2 X there exists a unique ¥ sequence (ln)n2N ⊂ K in such a way that ∑n=1 lnen converges to x. For every n 2 N, the coefficient functionals and the canonical projections are, respectively, defined by ∗ e : X ! K n (1) x 7! ln and pn : X ! spanfe1,..., eng n (2) x 7! ∑i=1 liei. ∗ For simplicity purposes, we convey that p0 is the null projection. Notice that pn(x) = e1 (x)e1 + ∗ ∗ ··· + en(x)en and pn(x) − pn−1(x) = en(x)en for all n 2 N and all x 2 X. As a consequence, if the Symmetry 2019, 11, 1026; doi:10.3390/sym11081026 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1026 2 of 12 coefficient functionals are all continuous, then the canonical projections are also continuous due to the vector character of the topology (see Lemma1). In this situation, we say that (en)n2N is a Schauder basis. In normed spaces, it is a trivial fact that the coefficient functionals are continuous if and only if the canonical projections are also continuous. However, the same fact in the topological vector space setting is not trivial at all. In Section2, we prove that the continuity of the canonical projections imply the continuity of the coefficient functionals in topological vector spaces over an absolutely valued division ring (Corollary1). We also show in this section that the existence of a pre-Schauder basis on a topological vector space over an absolutely valued division ring implies the existence of a dense Hausdorff subspace, which is precisely the linear span of the basis (Theorem1). We also construct explicitly non-Hausdorff topological vector spaces admitting Schauder bases (Theorem2). ∗ ⊆ × ∗ ∗( ) = In a topological vector space X, a biorthogonal system is a pair ei, ei i2I X X such that ei ej dij where dij is the Kronecker d. A biorthogonal system is said to be: • expanding if X = spanfei : i 2 Ig; • fundamental if X = spanfei : i 2 Ig; and ∗ w∗ ∗ • total if X = span fei : i 2 Ig. We refer the reader to [6] for a complete perspective on biorthogonal systems in real or complex ∗ normed spaces. If (en)n2N is a pre-Schauder basis in a topological vector space X, then (en, en)n2N is clearly a fundamental biorthogonal system. In Section2, we prove that, if X is Hausdorff and ∗ ∗ w -strongly torsionless, then (en, en)n2N is also total (Corollary1). A topological space is Hausdorff provided that for every two different points there can be found disjoint neighborhoods of the points. Let us recall now the concept of w∗-strongly torsionless. Let M be a topological (left) module over a topological ring R. Then, M∗ is a right R-module which can be endowed with the w∗-topology (the module topology on M∗ inherited from the product topology of RM). If A is a nonempty subset of M∗, > T ∗ then the preannihilator of A is defined by A := m∗2A ker(m ). The topological module M is said to be w∗-strongly torsionless if for every proper w∗-closed submodule N of M∗ and every m∗ 2 M∗ n N there exists m 2 N> n fm∗g>. In virtue of the Hahn–Banach Theorem, real or complex locally convex Hausdorff topological vector spaces are w∗-strongly torsionless. If A ⊆ M is nonempty, then the annihilator of A is defined as A? := fm∗ 2 M∗ : A ⊆ ker(m∗)g. Suppose now that X is a real or complex normed space admitting a pre-Schauder basis (en)n2N. The set V := fx 2 X : 8n 2 N kpn(x)k ≤ 1g is clearly absolutely convex. Since (pn(x))n2N is a convergent sequence to x, it is not hard to check that V is also absorbing. Thus, cl(V) is a barrel of X. If X is a Banach space, then it can be proved that V is closed for convex series and therefore int(V) = int(cl(V)). Since Banach spaces are barrelled, we conclude that int(V) = int(cl(V)) is not empty. This implies that all the canonical projections are continuous and the set fpn : n 2 Ng is bounded in B(X). In particular, the coefficient functionals are also continuous due to the norm properties. As a consequence, in Banach spaces, all pre-Schauder bases are Schauder bases and bc ((en)n2N) := supfkpnk : n 2 Ng is called the basis constant. This basis constant is key in order to renorm Banach spaces with Schauder bases to make those bases monotone. We refer the reader to [7–10] for recent developments on Schauder bases in Banach spaces. In Section3, we prove 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 (Theorem3). We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized (Theorem4 and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized (Corollary2)). 2. Impact of Pre-Schauder Bases on the Vector Topology First off, notice that a pre-Schauder basis on a topological vector space must be a linearly independent set. Indeed, if l1, ... , lk are scalars such that l1e1 + ··· + lkek = 0, then the sequence Symmetry 2019, 11, 1026 3 of 12 (gn)n2N ⊆ K defined by gi = li for 1 ≤ i ≤ k and gi = 0 for i ≥ k + 1 verifies that ∑n2N gnen converges to 0. Since the constant sequence equal to 0 also verifies the previous convergence, by uniqueness it must occur that gi = 0 for 1 ≤ i ≤ k. Recall that in a topological space X, the filter of neighborhoods of an element x 2 X is usually X denoted by Nx (or simply by Nx if there is no confusion with X). Recall also that the following conditions are trivially equivalent: • 2 \N = T y x : V2Nx V. • If (yi)i2I is a net in X converging to y, then it also converges to x. If X is a topological vector space, then it is not hard to show that y 2 \Nx if and only if x 2 \Ny. In fact, x + \Ny = \Nx+y for all x, y 2 X. On the other hand, in [11], it was shown that WX := \N0 is a closed vector subspace of X whose inherited topology is the trivial topology and which is topologically complemented with any of its linear complements. Any subspace Y of X verifying that Y \ WX = f0g is trivially Hausdorff. In particular, X is Hausdorff if and only if WX = f0g. Recall also that if M is a subset of a vector space X, then the series span of M is defined as ( ) sspan(M) := x 2 X : 9 (ln)n2N ⊆ K and (mn)n2N ⊆ M such that ∑ lnmn converges to x . n2N Notice that if X is a topological vector space with a pre-Schauder basis (en)n2N, then X = sspan ((en)n2N) = span ((en)n2N).
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