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2 results

  • Geometry of Spaces of Vector-Valued Harmonic Functions

  • Patrick N. Dowling, Zhibao Hu, Mark A. Smith
  • Journal:
    Canadian Journal of Mathematics / Volume 46 / Issue 2 / 01 April 1994
    Published online by Cambridge University Press:
    20 November 2018, pp. 274-283
    Print publication:
    01 April 1994
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  • It is shown that the space hp(D,X) has the Kadec-Klee property with respect to pointwise norm convergence in the Banach space X if and only if X has the Radon-Nikodym property and every point of the unit sphere of X is a denting point of the unit ball of X. In addition, it is shown that hp(D,X) is locally uniformly rotund if and only if X is locally uniformly rotund and has the Radon-Nikodym property.

  • Locally Uniformly Rotund Renorming and Injections Into c0(Γ)

  • G. Godefroy, S. Troyanski, J. Whitfield, V. Zizler
  • Journal:
    Canadian Mathematical Bulletin / Volume 27 / Issue 4 / 01 December 1984
    Published online by Cambridge University Press:
    20 November 2018, pp. 494-500
    Print publication:
    01 December 1984
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  • A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim |xn — x| = 0 for every xn, x ∈ X for which lim2|x|2 + 2 |xn|2-|xn+xn|2 = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c0(Γ) such that T* c*0(Γ) is norm dense in X*. This is the case e.g. if X* is weakly compactly generated (WCG).