Diestel, J.; Ruess, W. M.; Schachermayer, W. Weak compactness in \(L^ 1(\mu,X)\). (English) Zbl 0785.46037 Proc. Am. Math. Soc. 118, No. 2, 447-453 (1993). For any Banach space \(X\), let \(L^ 1(\mu,X)\) denote the space of Bochner integrable functions from a probability space \((\Omega,\Sigma,\mu)\) to \(X\) with the norm given by \(\| f\|_ 1= \int\| f\| d\mu\). A characterization of weak compactness in \(L^ 1(\mu,X)\) which generalizes a more restricted result of Ulger is proved. Corollaries which re-phrase the result in language reminiscent of the classical Eberlein-Smulian theorem on weak compactness in an arbitrary Banach space are also derived, as is an extension to order continuous Banach lattices having a weak unit. Reviewer: J.R.Holub (Blacksburg) Cited in 2 ReviewsCited in 82 Documents MSC: 46E40 Spaces of vector- and operator-valued functions 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46A50 Compactness in topological linear spaces; angelic spaces, etc. 46B25 Classical Banach spaces in the general theory 46B42 Banach lattices Keywords:space of Bochner integrable functions; weak compactness; Eberlein-Smulian theorem on weak compactness is an arbitrary Banach space; order continuous Banach lattices having a weak unit PDFBibTeX XMLCite \textit{J. Diestel} et al., Proc. Am. Math. Soc. 118, No. 2, 447--453 (1993; Zbl 0785.46037) Full Text: DOI