Abstract

Let  E \,E\, be an ideal of  Lo \,L^{\rm o}\, over a  σ\,\sigma-finite measure space  (Ω,Σ,μ)\,(\Omega, \Sigma, \mu), and let  (X,∥⋅∥X) \,(X, \norm \cdot \norm_X)\, be a real Banach space. Let  E(X) \,E(X)\, be a subspace of the space  Lo(X) \,L^{\rm o}(X)\, of  μ\,\mu-equivalence classes of all strongly  Σ\,\Sigma-measurable functions  f:  Ω⟶X \,f:\; \Omega\longrightarrow X\, and consisting of all those  f∈Lo(X) \,f\in L^{\rm o}(X)\, for which the scalar function  ∥f(⋅)∥X \,\norm f(\cdot) \norm_X\, belongs to  E\,E. Let  E(X)n∼ \,E(X)_n^{\sim}\, stand for the order continuous dual of  E(X)\,E(X). In this paper we characterize both conditionally  σ(E(X),I)\,\sigma(E(X),I)-compact and relatively  σ(E(X),I)\,\sigma(E(X), I)-sequentially compact subsets of  E(X) \,E(X)\, whenever  I \,I\, is an ideal of  E(X)n∼\,E(X)_n^{\sim}. As an application, we obtain a characterization of almost reflexivity and reflexivity of a Banach space  X \,X\, in terms of conditionally  σ(E(X),I)\,\sigma(E(X), I)-compact and relatively  σ(E(X),I)\,\sigma(E(X), I)-sequentially compact subsets of  E(X)\,E(X).

Author information

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

Marian Nowak

Faculty of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65–516 Zielona Góra, Poland

M.Nowak@wmie.uz.zgora.pl

Keywords

  • Vector-valued function spaces
  • Köthe-Bochner spaces
  • conditional weak compactness
  • weak sequential compactness
  • weak compactness
  • weak sequential completeness
  • almost reflexivity
  • reflexivity
  • absolute weak topologies

Mathematics Subject Classification

46E40; 46A50, 46A20, 46A25

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Suggested citation

M. Nowak. “Conditional and Relative Weak Compactness in Vector-Valued Function Spaces.” Journal of Convex Analysis 12 (2005), No. 2, 447–463. https://doi.org/10.68381/jca12031

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Published by Heldermann Verlag, 2005. Rights now held by Banach Press LLC.