Local uniform convexity and Kadec‐Klee type properties in K‐interpolation spaces I: General Theory

Abstract

We present a systematic study of the interpolation of local uniform convexity and Kadec‐Klee type properties in K‐interpolation spaces. Using properties of the K‐functional of J.Peetre, our approach is based on a detailed analysis of properties of a Banach couple and properties of a K‐interpolation functional which guarantee that a given K‐interpolation space is locally uniformly convex, or has a Kadec‐Klee property. A central motivation for our study lies in the observation that classical renorming theorems of Kadec and of Davis, Ghoussoub and Lindenstrauss have an interpolation nature. As a partiular by‐product of our study, we show that the theorem of Kadec itself, that each separable Banach space admits an equivalent locally uniformly convex norm, follows directly from our approach.