Functional analysis in asymmetric normed spaces Ştefan Çobzas

Format:

Book

Author/Creator:

Cobzaş, Ştefan

Series:

Frontiers in mathematics 1660-8046

Language:

English

Subjects (All):

Functional analysis.

Physical Description:

1 online resource

Place of Publication:

New York Springer 2013

Summary:

An asymmetric norm is a positive definite sublinear functional p on a real vector space X. The topology generated by the asymmetric norm p is translation invariant so that the addition is continuous, but the asymmetry of the norm implies that the multiplication by scalars is continuous only when restricted to non-negative entries in the first argument. The asymmetric dual of X, meaning the set of all real-valued upper semi-continuous linear functionals on X, is merely a convex cone in the vector space of all linear functionals on X. In spite of these differences, many results from classical functional analysis have their counterparts in the asymmetric case, by taking care of the interplay between the asymmetric norm p and its conjugate. Among the positive results one can mention: Hahn-Banach type theorems and separation results for convex sets, Krein-Milman type theorems, analogs of the fundamental principles - open mapping, closed graph and uniform boundedness theorems - an analog of the Schauder's theorem on the compactness of the conjugate mapping. Applications are given to best approximation problems and, as relevant examples, one considers normed lattices equipped with asymmetric norms and spaces of semi-Lipschitz functions on quasi-metric spaces. Since the basic topological tools come from quasi-metric spaces and quasi-uniform spaces, the first chapter of the book contains a detailed presentation of some basic results from the theory of these spaces. The focus is on results which are most used in functional analysis - completeness, compactness and Baire category - which drastically differ from those in metric or uniform spaces. The book is fairly self-contained, the prerequisites being the acquaintance with the basic results in topology and functional analysis, so it may be used for an introduction to the subject. Since new results, in the focus of current research, are also included, researchers in the area can use it as a reference text

Contents:

Introduction.-1. Quasi-metric and Quasi-uniform Spaces. 1.1. Topological properties of quasi-metric and quasi-uniform spaces

1.2. Completeness and compactness in quasi-metric and quasi-uniform spaces.-2. Asymmetric Functional Analysis

2.1. Continuous linear operators between asymmetric normed spaces

2.2. Hahn-Banach type theorems and the separation of convex sets

2.3. The fundamental principles

2.4. Weak topologies

2.5. Applications to best approximation

2.6. Spaces of semi-Lipschitz functions

Bibliography

Index

Notes:

Includes bibliographical references and index

Other Format:

Print version Çobzas, Ştefan. Functional analysis in asymmetric normed spaces

ISBN:

9783034804783

3034804784

OCLC:

823741715

Access Restriction:

Restricted for use by site license