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Introduction

  • Herbert Gross
Part of the Progress in Mathematics book series (PM, volume 1)

Abstract

No one would assert that finite dimensionality is an intrinsic feature of the concept of quadratic form. Yet, apart from a very small number of results (see References to Chapter XI) there has been, as far as we know, only Kaplansky’s 1950 paper on infinite dimensional spaces pointing our way, namely in the direction of a purely algebraic theory of quadratic forms on infinite dimensional vector spaces over “arbitrary” division rings. Such a theory would, naturally, leave aside the highly developed theory of Hilbert spaces and its relatives, Krein spaces, Pontrjagin spaces (see [2] for an orientation on these topics). Furthermore, when we speak of infinite dimensional geometric algebra we do not, in this book, mean discussion of the ramifications into geometry of hypotheses belonging to set theory nor, reversely, the study of axioms forced upon set theory by geometry. We simply mean that the (algebraic) dimension of the quadratic spaces is allowed to be infinite. Many problems of the finite dimensional setting remain perfectly meaningful and invite an investigation when the finiteness condition is removed. Our results show that it is possible to generalize, without rarefying, classical results from finite dimensional orthogonal geometry.

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References

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    E. Artin, Geometric Algebra. Interscience Publ. New York 1957.Google Scholar
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    J. Bognár, Indefinite inner product spaces. Ergebnisse Band 78, Springer, Berlin Heidelberg New York 1974.CrossRefGoogle Scholar
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    M. Eichler, Quadratische Formen und orthogonale Gruppen. Grundlehren Band 63, 2. Aufl., Springer, Berlin Heidelberg New York 1974.CrossRefGoogle Scholar
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    I. Kaplansky, Linear Algebra and Geometry. Allyn and Bacon, Boston 1969.Google Scholar
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    O. T. O’Meara, Introduction to quadratic forms. Grundlehren Band 177 Springer, Berlin Heidelberg New York 1974.Google Scholar
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    V. S. Varadarajan, Geometry of Quantum Theory, vol. 1. van Nostrand Princeton 1968.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • Herbert Gross
    • 1
  1. 1.Mathematisches InstitutUniversität ZürichZürichSwitzerland

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