ON THE EXTENSION OF k-NUC NORMS

Abstract

It is well-known that if Y is a closed subspace of a separable Banach space X; and if Y admits a locally uniformly rotund (LUR) norm, then this LUR norm on Y can be extended to a LUR norm on X: (See [4,5] and II.8 of [1].) In [2], Fabian used a new technique to extend the result to other forms of rotundity without requiring X to be separable. However in [2], the subpace Y has to be reflexive. The second-named author later showed that the condition that Y is reflexive could be removed. (See [8] and [9].) It is worth noting that the techniques used in the above three instances are completely different and each has its own strength. Fabian’s extension preserves moduli of convexity of power type. The extension given in [8] is simple and turns out to be very useful in many situations.