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2 results

  • DUAL DIFFERENTIATION SPACES

  • Part of
  • WARREN B. MOORS, NEŞET ÖZKAN TAN
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019
    Published online by Cambridge University Press:
    28 March 2019, pp. 467-472
    Print publication:
    June 2019
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  • We show that if $(X,\Vert \cdot \Vert )$ is a Banach space that admits an equivalent locally uniformly rotund norm and the set of all norm-attaining functionals is residual then the dual norm $\Vert \cdot \Vert ^{\ast }$ on $X^{\ast }$ is Fréchet at the points of a dense subset of $X^{\ast }$. This answers the main open problem in a paper by Guirao, Montesinos and Zizler [‘Remarks on the set of norm-attaining functionals and differentiability’, Studia Math.241 (2018), 71–86].

  • A dual differentiation space without an equivalent locally uniformly rotund norm

  • Part of
  • Petar S. Kenderov, Warren B. Moors
  • Journal:
    Journal of the Australian Mathematical Society / Volume 77 / Issue 3 / December 2004
    Published online by Cambridge University Press:
    09 April 2009, pp. 357-364
    Print publication:
    December 2004
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  • A Banach space (X, ∥ · ∥) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Fréchet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.