Abstract

We show that if μ is a probability measure and X is a Banach space, then the Lebesgue-Bochner space L1(μ,X)L^1(\mu,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)L^1(\mu,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.

Author information

Contact details are reproduced from the original publication and may be historical.

Marián Fabian

Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

fabian@math.cas.cz

Sebastián Lajara

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

sebastian.lajara@uclm.es

Keywords

  • Lebesgue-Bochner space
  • rotund norm
  • URED norm
  • LUR norm
  • MLUR norm
  • UR norm
  • Luxemburg norm
  • Orlicz function

Mathematics Subject Classification

46B03, 46B20, 46E40

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Suggested citation

M. Fabian, S. Lajara. “Rotund Renormings in Spaces of Bochner Integrable Functions.” Journal of Convex Analysis 22 (2015), No. 4, 1025–1039. https://doi.org/10.68381/jca22054

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.