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\(I\)-convexity and \(Q\)-convexity in Orlicz-Bochner function spaces equipped with the Luxemburg norm. (English) Zbl 1421.46015

Summary: We study \(I\)-convexity and \(Q\)-convexity, two geometric properties introduced by D. Amir and C. Franchetti [Trans. Am. Math. Soc. 282, 275–291 (1984; Zbl 0543.46007)]. We point out that a Banach space \(X\) has the weak fixed-point property when \(X\) is \(I\)-convex (or \(Q\)-convex) with a strongly bimonotone basis. By means of some characterizations of \(I\)-convexity and \(Q\)-convexity in Banach spaces, we obtain criteria for these two convexities in the Orlicz-Bochner function space \(L_{(M)}(\mu,X)\): that \(L_{(M)}(\mu,X)\) is \(I\)-convex (or \(Q\)-convex) if and only if \(L_{(M)}(\mu)\) is reflexive and \(X\) is \(I\)-convex (or \(Q\)-convex).

MSC:

46B20 Geometry and structure of normed linear spaces
46E40 Spaces of vector- and operator-valued functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0543.46007
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Full Text: DOI Euclid

References:

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