Generalized quasi-Banach sequence spaces and measures of noncompactness

Given 0 < s ≤ 1 and ψ an s-convex function, s – ψ -sequence spaces are introduced. Several quasi-Banach sequence spaces are thus characterized as a particular case of s – ψ -spaces. For these spaces, new measures of noncompactness are also defined, related to the Hausdorff measure of noncompactness. As an application, compact sets in s – ψ -interpolation spaces of a quasi-Banach couple are studied.

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.

Embedding functions and their role in interpolation theory

The embedding functions of an intermediate spaceAinto a Banach couple(A0,A1)are defined as its embedding constants into the couples(1αA0,1βA1),∀α,β>0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach couple to obtain new interpolation theorems.