On nearly uniformly convex and k-uniformly convex spaces

Abstract In this note we prove that every nearly uniformly convex space has normal structure and that K -uniformly convex spaces are super-reflexive. We recall 1 that a Banach space is said to be Kadec–Klee if whenever x n → x weakly and ∥ n ∥ = ∥ x ∥ = 1 for all n then ∥ x n − x ∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in 1. For the reader's convenience we recall them here.