Positively norming sets in Banach function spaces
Denote by \(E\) and \(E^{\ast }\) a Banach space and its dual, respectively. The set \(A\subset E^{\ast }\) is called norming if \(\inf_{x\in S\left( E\right) }\sup_{x^{\ast }\in A}\left| x^{\ast }\left( x\right) \right| >0\), where \(S\left( E\right) \) is the unit sphere of \(E\). For a Banach function space \(E=E\left( \mu \right) \), it is natural to consider norming sets in the sense of the Köthe dual \(E'\) of \(E\). Recall that \( E^{\ast }=E'\) if and only \(E\) is order continuous. Let \(B\left( E' \right) ^{+}=\left\{ x \in E':\left\| x\right\| _{E'}\leq 1\text{ and }x\geq 0\right\} \). We say that a set \(A\subset B\left( E'\right) ^{+}\) is positively \( \alpha \)-norming (for \(0<\alpha \leq 1\)) whenever \(\inf_{f\in S\left( E\right) }\sup_{g\in A}\int \left| f\right| gd\mu =\alpha\). A set \(A \) is positively norming if it is \(\alpha \)-norming for some \(0<\alpha \leq 1\). The authors prove that the existence of an order bounded positively norming set in \(B\left( E'\right) ^{+}\) is equivalent to the existence of a lattice copy of \(L^{1}\) in \(E\left( \mu \right) \) (provided that \(E\left( \mu \right) \) is order continuous and \(\mu \) is finite) (Theorem 3.2). They also show that \( E\left( \mu \right) \) is lattice isomorphic to a \(c_{0}\) (disjoint) sum of \( L^{1}\) spaces if and only if there is a positively norming set consisting of disjoint elements (under the assumption that \(E\left( \mu \right) \) is order continuous) (Theorem 3.3). They also introduce the notion of a \(p\)-convex \(1\)-norming set in order to prove that the existence of such a set implies \(p\)-concavity of \(E\left( \mu \right) \) (Corollary 3.7). They also prove that an order continuous Banach function space \(E\left( \mu \right) \) contains a band isomorphic to a space \(L^{1}\left( \upsilon \right) \), where \( \upsilon \) is absolutely continuous with respect to \(\mu \) if and only if there exists a positively norming set satisfying additional conditions (Theorem 3.12). In the next section, the authors show a version of the Kadec-Pełczyński dichotomy, for a subspace \(Y\) of an order continuous Banach function space \(E\left( \mu \right) \) with a weak unit, in terms of the measure of non-compactness of its positively norming sets. In the last section, the authors apply vector measures to show a procedure for defining positively norming sets.
scientific article; zbMATH DE number 6353675