Weak and norm convergence on the unit sphere (Q1801944)

Let \(X\) be a Banach space. \(X\) is said to have Kadec-Klee property ((KK)) if \(\{x_ n\}\) is a sequence of elements in \(X\) converging weakly to an \(x \in X\) such that \(\| x_ n \| \to \| x \|\), then \(\{x_ n\}\) converges to \(x\) in norm, and to have Kadec property ((K)) if the weak and the norm topology coincide on the unit sphere. In connection with a result of I. E. Leonard on property (KK) in Banach products of \(\ell^ p\) type [J. Math. Anal. Appl. 54, 245-265 (1976; Zbl 0343.46010)], the author obtains the following: For \(1 \leq p<\infty\) and a family \(\{X_ n\}\) of Banach spaces, let \(\ell^ p (X_ 1,X_ 2,\dots)\) be the Banach space defined by \(\{x=(x_ n):x_ n \in X_ n (n=1,2,\dots)\), \(\| x \|^ p_ p = \sum^ \infty_{n=1} \| x_ n \|^ p<\infty\}\) endowed with \(\| \cdot \|_ p\). Then \(\ell^ p (X_ 1,X_ 2,\dots)\) has property (KK) (resp. (K)) if and only if \(X_ n(n=1,2,\dots)\) has (KK) (resp. (K)). The author also establishes the stability of (KK) and (K) for the spaces introduced by \textit{R. Huff} [Rocky Mt. J. Math. 10, 743-749 (1980; Zbl 0505.46011)], that generalize ``Banach products'' for a countable quantity of spaces.