Spaceability of strict order integrability
type: Item
instance of: scholarly article
title: Spaceability of strict order integrability
zbMATH Open document ID: 1234.46028
author: L. Bernal-González
author: Manuel Ordóñez Cabrera
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-10-24T00:00:00Z
review text: A subset \(B\) of a topological vector space \(E\) is lineable (spaceable) if \(B\cup\{0\}\) contains an infinite dimensional (closed infinite-dimensional) linear subspace of \(E\). In the last decade the notions of lineability and spaceability were investigated by several authors in quite different contexts. The essence of these notions is the search for linear properties in subsets of vector spaces with a priori no linear structure. In this paper the authors obtain new results on the spaceability of subsets of \(L^{p}\) spaces, generalizing previous works of different authors. If \((X,\Sigma,\mu)\) is a measure space, consider the properties \[ \inf\left\{ \mu(A):A\in\Sigma,\mu(A)>0\right\} =0,\tag{\(\alpha\)} \] \[ \sup\left\{ \mu(A):A\in\Sigma,\mu(A)<\infty \right\} =\infty\tag{\(\beta\)} \] and, for \(p\in[1,\infty],\) define \[ L_{\text{l-strict}}^{p}:=L^{p}(\mu,X)\setminus\bigcup_{q\in[1,p)}L^{q}(\mu,X), \] \[ L_{\text{r-strict}}^{p}:=L^{p}(\mu,X)\setminus\bigcup_{q\in(p,\infty]}L^{q}(\mu,X), \] and \[ L_{\text{strict}}^{p}:=L^{p}(\mu,X)\setminus\bigcup_{q\in[1,\infty]\setminus\{p\}}L^{q}(\mu,X). \] The paper provides interesting characterizations of the spaceability of these classes in terms of the properties \((\alpha)\) and \((\beta).\) More precisely, (a) If \(p<\infty\), then \(L_{\text{r-strict}}^{p}\) is spaceable if and only if \((\alpha)\) holds. (b) If \(p>1\), then \(L_{\text{l-strict}}^{p}\) is spaceable if and only if \((\beta)\) holds. (c) If \(1<p<\infty\), then \(L_{\text{strict}}^{p}\) is spaceable if and only if both \((\alpha)\) and \((\beta)\) hold. In fact, if \(X\) is a nonempty set, the authors prove a quite general result about spaceability in subsets of Banach spaces of \(\mathbb{K}\)-valued functions on \(X\). In the final section, as an application of this general result, the authors obtain some spaceability results in spaces of continuous functions of bounded variation.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 28E99
Mathematics Subject Classification ID: 28A99
zbMATH DE Number: 5962610
zbMATH Keywords: lineability
zbMATH Keywords: spaceability
zbMATH Keywords: \(L_p\) spaces
zbMATH Keywords: bounded variation
reviewed by: Daniel Pellegrino
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.06.043
OpenAlex ID: W2006440786
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DOI: 10.1016/J.JMAA.2011.06.043
description: scientific article; zbMATH DE number 5962610