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Introduction to the Theory of Bases

  • Jürg T. Marti
Book

Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 18)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Jürg T. Marti
    Pages 1-17
  3. Jürg T. Marti
    Pages 18-27
  4. Jürg T. Marti
    Pages 28-54
  5. Jürg T. Marti
    Pages 69-78
  6. Jürg T. Marti
    Pages 79-85
  7. Jürg T. Marti
    Pages 86-97
  8. Jürg T. Marti
    Pages 98-112
  9. Back Matter
    Pages 130-151

About this book

Introduction

Since the publication of Banach's treatise on the theory of linear operators, the literature on the theory of bases in topological vector spaces has grown enormously. Much of this literature has for its origin a question raised in Banach's book, the question whether every sepa­ rable Banach space possesses a basis or not. The notion of a basis employed here is a generalization of that of a Hamel basis for a finite dimensional vector space. For a vector space X of infinite dimension, the concept of a basis is closely related to the convergence of the series which uniquely correspond to each point of X. Thus there are different types of bases for X, according to the topology imposed on X and the chosen type of convergence for the series. Although almost four decades have elapsed since Banach's query, the conjectured existence of a basis for every separable Banach space is not yet proved. On the other hand, no counter examples have been found to show the existence of a special Banach space having no basis. However, as a result of the apparent overconfidence of a group of mathematicians, who it is assumed tried to solve the problem, we have many elegant works which show the tight connection between the theory of bases and structure of linear spaces.

Keywords

Banach space Basis (Math.) Finite Topologischer Vektorraum Topology

Authors and affiliations

  • Jürg T. Marti
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-87140-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 1969
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-87142-9
  • Online ISBN 978-3-642-87140-5
  • Series Print ISSN 0081-3877
  • Buy this book on publisher's site
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