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Journal of Convex Analysis 22 (2015), No. 4, 1025--1039
Copyright Heldermann Verlag 2015

Rotund Renormings in Spaces of Bochner Integrable Functions

Marián Fabian
Institute of Mathematics, Czech Academy of Sciences, Zitná 25, 115 67 Praha 1, Czech Republic

Sebastián Lajara
Dep. de Matemáticas, Escuela de Ingenieros Industriales, Universidad de Castilla-La Mancha, Campus Universitario, 02071 Albacete, Spain

We show that if μ is a probability measure and X is a Banach space, then the Lebesgue-Bochner space L1(μ,X) admits an equivalent norm which is rotund (uniformly rotund in every direction, locally uniformly rotund, or midpoint locally uniformly rotund) if X does. We also prove that if X admits a uniformly rotund norm, then the space L1(μ,X) has an equivalent norm whose restriction to every reflexive subspace is uniformly rotund. This is done via the Luxemburg norm associated to a suitable Orlicz function.

Keywords: Lebesgue-Bochner space, rotund norm, URED norm, LUR norm, MLUR norm, UR norm, Luxemburg norm, Orlicz function.

MSC: 46B03, 46B20, 46E40

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