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26077 results in Abstract analysis

Page 1143 of 1304

  • Existence and stability in convection in porous media

  • Part of
  • Moses A. Boudourides
  • Journal:
    Mathematika / Volume 34 / Issue 1 / June 1987
    Published online by Cambridge University Press:
    26 February 2010, pp. 39-52
    Print publication:
    June 1987

  • Consider an impermeable container Ω in ℝ3 filled with a porous material saturated with a Boussinesq fluid. The boundary ∂Ω of Ω consists of two parts Γ1 and Γ2, i. e., ∂Ω = Γ1 ⋃ Γ2. Γ1 is the intersection of the horizontal planes at z = 0 and z = 1 with a vertical cylinder of arbitrary cross section G (a bounded smooth domain in ℝ2), i.e., Γ1 = G × {0, 1}. Γ2 is the sidewall of the vertical cylinder between the planes z = 0 and z = l, i.e., Γ2Γ = ∂G × [0, 1].

  • Topological spaces with dense subspaces that are homeomorphic to complete metric spaces and the classification of C(K) Banach spaces

  • Part of
  • Charles Stegall
  • Journal:
    Mathematika / Volume 34 / Issue 1 / June 1987
    Published online by Cambridge University Press:
    26 February 2010, pp. 101-107
    Print publication:
    June 1987

  • In [S1] we introduced and in [S2, S3, S4] developed a class of topological spaces that is useful in the study of the classification of Banach spaces and Gateaux differentiation of functions defined in Banach spaces. The class C may be most succinctly defined in the following way: a Hausdorff space T is in C if any upper semicontinuous compact valued map (usco) that is minimal and defined on a Baire space B with values in T must be point valued on a dense Gδ subset of B. This definition conceals many interesting properties of the family C. See [S2] for a discussion of the various definitions. Our main result here is that if X is a Banach space such that the dual space X* in the weak* topology is in C and K is any weak* compact subset of X* then the extreme points of K contain a dense, necessarily Gδ, subset homeomorphic to a complete metric space. In [S4] we studied the class K of κ-analytic spaces in C. Here we shall show that many elements of K contain dense subsets homeomorphic to complete metric spaces. It is easy to see that C contains all metric spaces and it is proved in [S4] that analytic spaces are in K. We obtain a number of topological results that may be of independent interest. We close with a discussion of various examples that show the interaction of these ideas between functional analysis and topology

Page 1143 of 1304