Extremal characterizations of reflexive spaces

Assume that a Banach space has a Fréchet differentiable and locally uniformly convex norm. We show that the reflexive property of the Banach space is not only sufficient, but also a necessary condition for the fulfillment of the proximal extremal principle in nonsmooth analysis.

Local uniform convexity and Kadec-Klee type properties inK-interpolation spaces I: General Theory

We present a systematic study of the interpolation of local uniform convexity and Kadec-Klee type properties inK-interpolation spaces. Using properties of theK-functional of J.Peetre, our approach is based on a detailed analysis of properties of a Banach couple and properties of aK-interpolation functional which guarantee that a givenK-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.