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

The uniformly Kadec-Klee property in Kothe-Bochner sequence spaces E(X), where E is a Kothe 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 Kothe function space. As a corollary we get criteria for E(X) to be nearly uniformly convex.