\(H\)-points in Köthe-Bochner spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 0992.46030
author: Dorota Krassowska
publication date: 2001-08-12T00:00:00Z
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1633171
zbMATH Keywords: \(H\)-points
zbMATH Keywords: Kadec-Klee property
zbMATH Keywords: Köthe function Banach space
zbMATH Keywords: order continuous
zbMATH Keywords: topological dual
zbMATH Keywords: Köthe-Bochner space
zbMATH Keywords: extreme point in the unit ball
MaRDI profile type: Publication
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title: \(H\)-points in Köthe-Bochner spaces
published in: Annales Societatis Mathematicae Polonae. Seria I. Commentationes Mathematicae
review text: Let \((T,\Sigma,\mu)\) be a measure space with a complete, \(\sigma\)-finite and non-atomic measure \(\mu\). Let \(E\) be a Köthe function Banach space on this measure space which is order continuous and such that its topological dual \(E'\) is also order continuous. Let \(X\) be a real Banach space and let \(E(X)\) be the Köthe-Bochner space. If \(f\in E(X)\) is an element of norm 1 such that every sequence \((g_n)_n\) of elements of norm 1 weakly converging to \(f\) in \(E(X)\) is also norm convergent to \(f\) in \(E(X)\) (i.e. \(f\) is an \(H\)-point in \(E(X)\)), and such that \(\|f\|\) is an extreme point in the unit ball of \(E\), then \(f\) is an extreme point in the unit ball of \(E(X)\). This main result of the present article is an extension of a theorem of \textit{Z. Hu} and \textit{B. L. Lin} [Ill. J. Math. 37, No. 2, 329-347 (1993; Zbl 0839.46010)], and permits the author to improve a theorem of \textit{C. Castain} and \textit{R. Pluciennik} [C. R. Acad. Sci. Paris, Sér. I 319, No. 11, 1159-1163 (1994; Zbl 0824.46042)].
reviewed by: José Bonet
description: scientific article; zbMATH DE number 1633171