
Journal of Convex Analysis 22 (2015), No. 4, 10251039 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 fabian@math.cas.cz Sebastián Lajara Dep. de Matemáticas, Escuela de Ingenieros Industriales, Universidad de CastillaLa Mancha, Campus Universitario, 02071 Albacete, Spain sebastian.lajara@uclm.es We show that if μ is a probability measure and X is a Banach space, then the LebesgueBochner space L^{1}(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 L^{1}(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: LebesgueBochner space, rotund norm, URED norm, LUR norm, MLUR norm, UR norm, Luxemburg norm, Orlicz function. MSC: 46B03, 46B20, 46E40 [ Fulltextpdf (141 KB)] for subscribers only. 