Abstract

Let (X,∥.∥X)(X, \|.\|_X) be an order-continuous Banach ideal space over a σ-finite measure space (Ω, Σ, μ) and E a Banach space. We prove that a function f of the vector Banach ideal space X(E) is a denting point of the unit ball of X(E) if and only if: (i) the modulus function ∣f∣:t⟼∥f(t)∥|f|: t \longmapsto \|f(t)\| is a denting point of the unit ball of X and (ii) f(t)/∥f(t)∥f(t)/\|f(t)\| is a denting point of the unit ball of E for almost all t in supp(f). This gives an answer to the open problem raised in a paper of Castaing and Pluciennik

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H. Benabdellah

Dép. de Mathématiques, Faculté des Sciences, Semlalia, B.P. S15, Marrakech, Marocco

References

  1. H. Benabdellah: Contribution aux problèmes de convergence fort-faible, à la géometrie des espaces de Banach et aux inclusions différentielles, Thèse de Doctorat, Univ. Montpellier II, 1991.Google Scholar
  2. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin et al., Lect. Notes Math. 580, 1977.https://doi.org/10.1007/BFb0087685Google Scholar
  3. L. Kantorovitch, G. Akilov: Analyse Fonctionnelle: Tome 1, Editions Mir, Moscow, 1981.Google Scholar
  4. L. Lindenstrauss, L. Tzafriri: Classical Banach spaces II, Functions Spaces, Ergeb. Math. ihrer Grenzgeb. 97, Springer-Verlag, Berlin et al., 1979.https://doi.org/10.1007/978-3-662-35347-9Google Scholar
  5. B-L. Lin, P-K. Lin, S. L. Troyanski: Characterizations of denting points, Proc. Amer. Math. Soc. 102(3) (1988) 526-528.https://doi.org/10.1090/S0002-9939-1988-0928972-1Google Scholar
  6. M. Väth: Ideal Spaces, Lecture Notes in Math. 1664, Springer-Verlag, Berlin et al., 1997.https://doi.org/10.1007/BFb0093548Google Scholar
  7. P. P. Zabrejko: Ideal spaces of functions I, Vestnik Jarosl. Univ. (1974) 12-52 (in Russian).Google Scholar

Suggested citation

H. Benabdellah. “Denting Points in Bochner Banach Ideal Spaces X(E).” Journal of Convex Analysis 6 (1999), No. 1, 183–194. https://doi.org/10.68381/jca06011

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Published by Heldermann Verlag, 1999. Rights now held by Banach Press LLC.