Uniform Kadec-Klee property and nearly uniform convexity in Köthe-Bochner sequence spaces

Kolwicz, Paweł

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 221-235

Résumé

The uniformly Kadec-Klee property in Köthe-Bochner sequence spaces $E(X)$, where $E$ is a Köthe sequence space and $X$ is an arbitrary separable Banach space, is studied. Namely, the question of whether or not this geometric property lifts from $X$ and $E$ to $E(X)$ is examined. It is settled affirmatively in contrast to the case when $E$ is a Köthe function space. As a corollary we get criteria for $E(X)$ to be nearly uniformly convex.