\(H\)-points in Köthe-Bochner spaces (Q2733905)

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)].