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The Isometric Theory of Classical Banach Spaces

  • Book
  • © 1974

Overview

Authors:
  1. H. Elton Lacey

    1. Department of Mathematics, University of Texas at Austin, Austin, USA

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 208)

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Table of contents (7 chapters)

  1. Front Matter

    Pages I-X
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  2. Partially Ordered Banach Spaces

    • H. Elton Lacey
    Pages 1-26

  3. Some Aspects of Topology and Regular Borel Measures

    • H. Elton Lacey
    Pages 27-56

  4. Characterizations of Banach Spaces of Continuous Functions

    • H. Elton Lacey
    Pages 57-100

  5. Classical Sequence Spaces

    • H. Elton Lacey
    Pages 101-116

  6. Representation Theorems for Spaces of the Type L p (T,ฮฃ,ฮผ,โ„‚)

    • H. Elton Lacey
    Pages 117-139

  7. Characterizations of Abstract M and L p Spaces

    • H. Elton Lacey
    Pages 140-181

  8. L1-Predual Spaces

    • H. Elton Lacey
    Pages 182-248

  9. Back Matter

    Pages 249-272
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Keywords

About this book

The purpose of this book is to present the main structure theorems in the isometric theory of classical Banach spaces. Elements of general topology, measure theory, and Banach spaces are assumed to be familiar to the reader. A classical Banach space is a Banach space X whose dual space is linearly isometric to Lp(j1, IR) (or Lp(j1, CC) in the complex case) for some measure j1 and some 1 ~ p ~ 00. If 1 < p < 00, then it is well known that X=L (j1,IR) where 1/p+1/q=1 and if p=oo, then X=L (v,lR) for q j some measure v. Thus, the only case where a space is obtained which is not truly classical is when p = 1. This class of spaces is known as L - 1 predual spaces since their duals are L type. It includes some well known j subclasses such as spaces of the type C(T, IR) for T a compact Hausdorff space and abstract M spaces. The structure theorems concern necessary and sufficient conditions that a general Banach space is linearly isometric to a classical Banach space. They are framed in terms of conditions on the norm of the space X, conditions on the dual space X*, and on (finite dimensional) subspaces of X. Since most of these spaces are Banach lattices and Banach algebras, characterizations among theses classes are also given.

Authors and Affiliations

  • Department of Mathematics, University of Texas at Austin, Austin, USA

    H. Elton Lacey

Bibliographic Information

  • Book Title: The Isometric Theory of Classical Banach Spaces

  • Authors: H. Elton Lacey

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-642-65762-7

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin ยท Heidelberg 1974

  • Softcover ISBN: 978-3-642-65764-1Published: 07 December 2011

  • eBook ISBN: 978-3-642-65762-7Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: X, 272

  • Topics: Analysis

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