Spaceability of strict order integrability
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.
scientific article; zbMATH DE number 5962610
scientific article