Geometric properties of some types of locally convex spaces

Abstract

It is well known that a normed space E is uniformly convex (smooth) normed space if and only if its dual E∗ is uniformly smooth (convex). We extend these geometric properties to seminormed spaces and then introduce definitions of uniformly convex (smooth) countably seminormed spaces. A new vision of the completion of countably seminormed space was helpful in our task. We get some fundamental links between Lindenstrauss duality formulas. A duality property between uniform convexity and uniform smoothness of countably seminormed space is also given. Also we give a definition of countably normed space associated with countably seminormed space with compatible seminorms and a definition of Metric projection in a countably seminormed space. This M. Sc. thesis is organized as follows: 1. Introduction, we show the importance of locally convex spaces and we give a general view of what we have been done in this thesis. 2. In chapter #1, we give a summary of topolpgical spaces [1], a directed system [1], topological vector spaces [2], Hausdorff topological vector Spaces [2] and quotient topological vector spaces [2] almost of the details needed in this thesis. 3. In chapter #2, we study locally convex spaces [2], seminorms [2] and metrizable topological vector spaces [2]. 4. In chapter #3, we study completion of different spaces (Metric space [5], normed space by using the technique of associated