Quadratic Forms in Infinite Dimensional Vector Spaces

1. Front Matter

   Pages N2-XII

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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 414-421

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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 (" NO-forms'''). Certain among the results 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 N-dimensional O spaces ideally serves the purpose. First, these spaces show a large number 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 NO it is easy to see, in a given case, wheth­ er topological language is appropriate or not).

• algebra
• Division
• Finite
• language
• linear algebra
• proof
• quadratic form
• recursion
• ring
• time
• Vector space

• Mathematisches Institut, Universität Zürich, Zürich, Switzerland

  Herbert Gross

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