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A notion of Orlicz spaces for vector valued functions. (English) Zbl 1099.46021

Summary: The notion of the Orlicz space is generalized to spaces of Banach space valued functions. A well-known generalization is based on \(N\)-functions of a real variable. We consider a more general setting based on spaces generated by convex functions defined on a Banach space. We investigate structural properties of these spaces, such as the role of the delta-growth conditions, separability, the closure of \(\mathcal L^{\infty }\), and representations of the dual space.

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
46B10 Duality and reflexivity in normed linear and Banach spaces
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