Geometric properties on non-complete spaces

Abstract

The purpose of this paper is to study certain geometrical properties for non-complete normed spaces. We show the existence of a non-rotund Banach space with a rotund dense maximal subspace. As a consequence, we prove that every separable Banach space can be renormed to be non-rotund and to contain a dense maximal rotund subspace. We then construct a non-smooth Banach space with a dense maximal smooth subspace. We also study the Krein-Milman property on non-complete normed spaces and provide a sufficient condition for an infinite dimensional Banach space to have an infinite dimensional, separable quotient.