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Remark on norm compactness in \(L^p(\mu,X)\). (English) Zbl 1471.46041

Summary: We prove a compactness criterion in \(L^p(\mu,X)\): a subset of \(L^p(\mu,X)\) is relatively norm compact iff the set of integrals of its functions over any measurable set is relatively norm compact, it satisfies the Fréchet oscillation restriction condition and it is \(p\)-uniformly integrable. The proof is elementary.

MSC:

46E40 Spaces of vector- and operator-valued functions
46B50 Compactness in Banach (or normed) spaces
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