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Introduction to the Theory of Bases

  • Book
  • © 1969

Overview

Authors:
  1. Jürg T. Marti

    1. Department of Mathematics, University of Illinois, Urbana, USA

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

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-XII
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  2. Linear Transformations

    • Jürg T. Marti
    Pages 1-17

  3. Convergence of Series in Banach Spaces

    • Jürg T. Marti
    Pages 18-27

  4. Bases for Banach Spaces

    • Jürg T. Marti
    Pages 28-54

  5. Orthogonality, Projections and Equivalent Bases

    • Jürg T. Marti
    Pages 55-68

  6. Bases and Structure of the Space

    • Jürg T. Marti
    Pages 69-78

  7. Bases for Hilbert Spaces

    • Jürg T. Marti
    Pages 79-85

  8. Decompositions

    • Jürg T. Marti
    Pages 86-97

  9. Applications to the Theory of Banach Algebras

    • Jürg T. Marti
    Pages 98-112

  10. Some Results on Generalized Bases for Linear Topological Spaces

    • Jürg T. Marti
    Pages 113-129

  11. Back Matter

    Pages 130-151
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, University of Illinois, Urbana, USA

    Jürg T. Marti

Bibliographic Information

  • Book Title: Introduction to the Theory of Bases

  • Authors: Jürg T. Marti

  • Series Title: Springer Tracts in Natural Philosophy

  • DOI: https://doi.org/10.1007/978-3-642-87140-5

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1969

  • Softcover ISBN: 978-3-642-87142-9Published: 18 April 2014

  • eBook ISBN: 978-3-642-87140-5Published: 13 March 2013

  • Series ISSN: 0081-3877

  • Edition Number: 1

  • Number of Pages: XII, 151

  • Topics: Functional Analysis

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