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Weak compactness in \(L^ 1(\mu,X)\). (English) Zbl 0785.46037

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.

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
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