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Locally solid topologies on vector valued function spaces. (English) Zbl 0912.46028
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 W. Wnuk [Bull. Pol. Acad. Sci., Math. 34, 413-416 (1986; Zbl 0605.46008)] and M. Novak [ibid. 32, 439-445 (1984; Zbl 0584.46018)].

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
46A80 Modular spaces
46A40 Ordered topological linear spaces, vector lattices
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