Weakly precompact subsets of L₁(μ,X)

• Autorzy: Ioana Ghenciu
• Języki publikacji: EN

Abstrakty

EN

Let (Ω,Σ,μ) be a probability space, X a Banach space, and L₁(μ,X) the Banach space of Bochner integrable functions f:Ω → X. Let W = {f ∈ L₁(μ,X): for a.e. ω ∈ Ω, ||f(ω)|| ≤ 1}. In this paper we characterize the weakly precompact subsets of L₁(μ,X). We prove that a bounded subset A of L₁(μ,X) is weakly precompact if and only if A is uniformly integrable and for any sequence (fₙ) in A, there exists a sequence (gₙ) with $gₙ ∈ co{f_i: i ≥ n}$ for each n such that for a.e. ω ∈ Ω, the sequence (gₙ(ω)) is weakly Cauchy in X. We also prove that if A is a bounded subset of L₁(μ,X), then A is weakly precompact if and only if for every ϵ >0, there exist a positive integer N and a weakly precompact subset H of NW such that A ⊆ H + ϵB(0), where B(0) is the unit ball of L₁(μ,X).

Kategorie tematyczne

• 46E40: Spaces of vector- and operator-valued functions
• 46B20: Geometry and structure of normed linear spaces
• 28B05: Vector-valued set functions, measures and integrals
• 46G10: Vector-valued measures and integration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2012
• Tom: 129
• Numer: 1
• Strony: 133-143

Daty

wydano

2012

Twórcy

autor

Ioana Ghenciu

• Mathematics Department, University of Wisconsin-River Falls, River Falls, WI 54022, U.S.A.

Identyfikatory

DOI

10.4064/cm129-1-10