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On weak convergence implying strong convergence under extremal conditions. (English) Zbl 0768.46013

T. Rzeżuchowski [Bull. Aust. Math. Soc. 39, No. 2, 201-214 (1989; Zbl 0651.28007)] showed that under a certain extremal subset condition the weak convergence in an \(L^ 1\)-space does imply the strong one. The author has improved upon and extended Rzeżuchowski’s result to the infinite-dimensional case by utilizing the theory of Young measures. His theorem also yields the main result of [same author, ibid. 33, 363-368 (1986; Zbl 0579.46018)].
Reviewer: T.Kubiak (Poznań)

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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