Differentiable and rough norms

Abstract

The principal question discussed in this dissertation is the problem of characterizing the existence of admissible Frechet differentiable norms on Banach spaces. In the first chapter the basic concepts of normed linear spaces are introduced and a summary of differential calculus on Banach spaces is given. The following three chapters of the paper are concerned with the existence of an admissible Frechet differentiable norm on a separable Banach space. A construction of such a norm is given for a separable space which has a separable dual. Also, it is shown that if such a norm exists on a Banach space, then the density character of the space equals the density character of its dual. In Chapter V, it is shown that if the density characters of a space and its dual are not equal, then the space admits a rough norm. As a consequence of this, there is no Frechet differentiable function on this space with bounded non-empty support. This implies that the space does not admit a Frechet differentiable norm.