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Conditional weak compactness in vector-valued function spaces

Author: Marian Nowak
Journal: Proc. Amer. Math. Soc. 129 (2001), 2947-2953
MSC (2000): Primary 46B25, 46E40
Published electronically: April 17, 2001
MathSciNet review: 1840098
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Let $E$ be an ideal of $L^{0}$ over a $\sigma $-finite measure space $(\Omega ,\Sigma ,\mu )$ and let $E^{\prime }$ be the Köthe dual of $E$ with $\hbox {supp}\,E^{\prime }=\Omega $. Let $(X,\Vert\cdot \Vert _{X})$ be a real Banach space, and $X^{*}$ the topological dual of $X$. Let $E(X)$ be a subspace of the space $L^{0}(X)$ of equivalence classes of strongly measurable functions $f\colon \, \Omega \to X$ and consisting of all those $f\in L^{0}(X)$ for which the scalar function $\Vert f(\cdot )\Vert _{X}$ belongs to $E$. For a subset $H$ of $E(X)$ for which the set $\{\Vert f(\cdot )\Vert _{X}\colon \, f\in H\}$ is $\sigma (E,E^{\prime })$-bounded the following statement is equivalent to conditional $\sigma (E(X),E^{\prime }(X^{*}))$-compactness: the set $\{\Vert f(\cdot )\Vert _{X}\colon \, f\in H\}$ is conditionally $\sigma (E,E^{\prime })$-compact and $\{\int _{A} f(\omega )d\mu \colon \, f\in H\}$ is a conditionally weakly compact subset of $X$ for each $A\in \Sigma $, $\mu(A)<\infty$ with $\chi _{A}\in E^{\prime }$. Applications to Orlicz-Bochner spaces are given.

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Additional Information

Marian Nowak
Affiliation: Institute of Mathematics, T. Kotarbiński Pedagogical University, Pl. Słowiański 9, 65–069 Zielona Góra, Poland

Keywords: Conditional weak compactness, vector valued function spaces
Received by editor(s): July 6, 1998
Received by editor(s) in revised form: February 14, 2000
Published electronically: April 17, 2001
Communicated by: Dale Alspach
Article copyright: © Copyright 2001 American Mathematical Society

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