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Quadratic Forms in Infinite Dimensional Vector Spaces

  • Book
  • © 1979

Overview

Authors:
  1. Herbert Gross

    1. Mathematisches Institut, Universität Zürich, Zürich, Switzerland

Part of the book series: Progress in Mathematics (PM, volume 1)

  • 5127 Accesses

  • 9 Citations

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About this book

For about a decade I have made an effort to study quadratic forms in infinite dimensional vector spaces over arbitrary division rings. Here we present in a systematic fashion half of the results found du­ ring this period, to wit, the results on denumerably infinite spaces (" ~O- forms") . Certain among the resul ts included here had of course been published at the time when they were found, others appear for the first time (the case, for example, in Chapters IX, X, XII where I in­ clude results contained in the Ph.D.theses by my students w. Allenspach, L. Brand, U. Schneider, M. Studer). If one wants to give an introduction to the geometric algebra of infinite dimensional quadratic spaces, a discussion of ~ -dimensional 0 spaces ideally serves the purpose. First, these spaces show a large nurober of phenomena typical of infinite dimensional spaces. Second, most proofs can be done by recursion which resembles the familiar pro­ cedure by induction in the finite dimensional Situation. Third, the student acquires a good feeling for the linear algebra in infinite di­ mensions because it is impossible to camouflage problems by topological expedients (in dimension ~O it is easy to see, in a given case, wheth­ er topological language is appropriate or not) .

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Table of contents (17 chapters)

  1. Front Matter

    Pages N1-XII
    Download chapter PDF

  2. Introduction

    • Herbert Gross
    Pages 1-3

  3. Fundamentals on Sesquilinear Forms

    • Herbert Gross
    Pages 4-60

  4. Diagonalization of ℵ0-Forms

    • Herbert Gross
    Pages 61-95

  5. Witt Decompositions for Hermitean ℵo-Forms

    • Herbert Gross
    Pages 96-109

  6. Isomorphisms between Lattices of Linear Subspaces Which are Induced by Isometries

    • Herbert Gross
    Pages 110-126

  7. Subspaces in Trace-Valued Spaces with Many Isotropic Vectors

    • Herbert Gross
    Pages 127-135

  8. Orthogonal and Symplectic Separation

    • Herbert Gross
    Pages 136-150

  9. Classification of Hermitean Forms in Characteristic 2

    • Herbert Gross
    Pages 151-168

  10. Subspaces in Non-Trace-Valued Spaces

    • Herbert Gross
    Pages 169-201

  11. Involutions in Hermitean Spaces in Characteristic Two

    • Herbert Gross
    Pages 202-224

  12. Extension of Isometries

    • Herbert Gross
    Pages 225-252

  13. Classification of Forms over Ordered Fields

    • Herbert Gross
    Pages 253-268

  14. Classification of Subspaces in Spaces with Definite Forms

    • Herbert Gross
    Pages 269-327

  15. Classification of ⊥-Dense Subspaces with Definite Forms

    • Herbert Gross
    Pages 328-353

  16. Quadratic Forms

    • Herbert Gross
    Pages 354-374

  17. Witts Theorem in Finite Dimensions

    • Herbert Gross
    Pages 375-386

  18. Arfs Theorem in Dimension ℵ0

    • Herbert Gross
    Pages 387-412

  19. Back Matter

    Pages 414-419
    Download chapter PDF
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Authors and Affiliations

  • Mathematisches Institut, Universität Zürich, Zürich, Switzerland

    Herbert Gross

Accessibility Information

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Bibliographic Information

  • Book Title: Quadratic Forms in Infinite Dimensional Vector Spaces

  • Authors: Herbert Gross

  • Series Title: Progress in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-1454-8

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1979

  • Softcover ISBN: 978-1-4757-1456-2Published: 17 September 2013

  • eBook ISBN: 978-1-4757-1454-8Published: 11 November 2013

  • Series ISSN: 0743-1643

  • Series E-ISSN: 2296-505X

  • Edition Number: 1

  • Number of Pages: XII, 419

  • Number of Illustrations: 2 b/w illustrations

  • Topics: Analysis, Mathematics, general

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