Locally solid topologies on vector valued function spaces
Let \(M\) denote the set of strongly measurable functions with values in a Banach space. A set \(E\subseteq M\) is called solid if \(y\in E\) and \(\| x(t)\|\leq\| y(t)\|\) (a.e.) implies \(x\in E\). If \(E\) is a topological space and \(0\) has a base of solid neighborhoods, the topology is called locally solid. In this paper, locally solid topologies are characterized. Moreover, many useful properties which are known in the normed case (for ideal spaces/Köthe spaces) are studied, e.g.: Continuity of the embedding of \(E\) into \(M\); analogues of the convergence theorems of Lebesgue and Vitali; the finest topology in Orlicz-Bochner spaces for which such theorems hold is characterized. The proofs reduce the problems to the scalar case which was studied in e.g. by \textit{W. Wnuk} [Bull. Pol. Acad. Sci., Math. 34, 413-416 (1986; Zbl 0605.46008)] and \textit{M. Novak} [ibid. 32, 439-445 (1984; Zbl 0584.46018)].
strongly measurable functions with values in a Banach space
convergence theorems of Lebesgue and Vitali
finest topology in Orlicz-Bochner spaces
scientific article; zbMATH DE number 1186790