Skip to main content
SpringerLink
Log in

Fundamentals on Sesquilinear Forms

  • Chapter
Quadratic Forms in Infinite Dimensional Vector Spaces

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

  • 275 Accesses

Abstract

Chapter I contains some of the basic concepts and facts upon which subsequent chapters are built. The reader will find the terminology and notations that are used throughout the text. A number of fundamental definitions have been inserted in later chapters; whenever it had been possible to introduce a concept right where it is needed without interrupting the flow of ideas we have postponed its introduction.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References to Chapter I

  1. A.A. Albert, Structure of Algebras. (3rd print of rev.ed.) AMS Coll Publ. XXIV, New York 1968.

    Google Scholar 

  2. R. Baer, Linear Algebra and Projective Geometry. Academic Press, New York 1952.

    MATH  Google Scholar 

  3. W. Baur and H. Gross, Strange inner product spaces. Comment. Math. Heiv. 52 (1977) 491–495.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Birkhoff, Lattice Theory. AMS Coll Publ. XXV (3rd ed,2nd print) Providence, R. I. 1973.

    Google Scholar 

  5. G. Birkhoff, and J. v. Neumann, The logic of quantum mechanics. Ann. of Math. 37 (1936) 823–843.

    Article  MathSciNet  Google Scholar 

  6. N. Bourbaki, Formes sesquilinéaires et formes quadratiques. Hermann Paris 1959.

    Google Scholar 

  7. J. Dieudonné, On the structure of unitary groups. Trans. Amer. Math. Soc. 72 (1952) 367–385.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Dieudonné, On the structure of unitary groups II. Amer. J. Math. 75 (1953) 665–678.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Dieudonné, La géometrie des groupes classiques, 3ième éd. Ergebnisse der Mathematik, Heft 5, Springer Berlin, Heidelberg 1971.

    Google Scholar 

  10. H.R. Fischer and H. Gross, Quadratic Forms and Linear Topologies I. Math. Ann. 157 (1964) 296–325.

    Article  MathSciNet  MATH  Google Scholar 

  11. H.R. Fischer and H. Gross, Tensorprodukte linearer Topologien (Quadratische Formen und lineare Topologien III). Math. Ann. 160 (1965) 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Frapolli, Generalizzazione di un teorema di H.A. Keller sulla modularità del reticolo dei sottospazi ortogonalmente chiusi di uno spazio sesquilineare. Masters Thesis, Univ. of Zurich 1975. (This concerns some technicalities when char k = 2; in [25] it was assumed that char k + 2)

    Google Scholar 

  13. H. Gross, On a representation theorem for AC-lattices. Mimeographed notes 1974.

    Google Scholar 

  14. H. Gross, Linearly topologized spaces without continuous bases (Quadratic forms and linear topologies V). Math. Ann. 194 (1971) 313–315.

    Google Scholar 

  15. H. Gross, and H.A. Keller, On the definition of Hilbert space. Manuscripta math. 23 (1977) 67–90.

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Gross, and V.H. Miller, Continuous forms in infinite dimensional spaces (Quadratic forms and linear topologies IV). Comment. Math. Helv. 42 (1967) 132–170.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Gross, and E. Ogg, On completions. Ann. Acad. Sci. Fenn. Ser. A. I 584 (1974) 1–19.

    MathSciNet  Google Scholar 

  18. H. Gross, and E. Ogg,, Quadratic spaces with few isometries. Comment. Math. Hell/. 48 (1973) 511–519.

    Article  MATH  Google Scholar 

  19. O. Hamara, On the structure of the orthogonal group. Math. Scand. 21 (1967) 219–232.

    MathSciNet  MATH  Google Scholar 

  20. O. Hamara, Quadratic forms on linearly topologized vector spaces. Portugal. Math. 27 (1968) 15–30.

    MathSciNet  MATH  Google Scholar 

  21. I.N. Herstein, On a Theorem of Albert. Scripta Mathematica XXIX (1973) 391–394.

    Google Scholar 

  22. I. Kaplansky, Forms in infinite dimensional spaces. Anais da Academia Brasileira de Ciencias 22 (1950) 1–17.

    MathSciNet  Google Scholar 

  23. I. Kaplansky,, Linear Algebra and Geometry. Allyn and Bacon, Boston 1969.

    MATH  Google Scholar 

  24. H.A. Keller, Stetigkeitsfragen bei lineartopologischen Cliffordalgebren. Ph.D. Thesis, University of Zurich 1971.

    Google Scholar 

  25. H.A. Keller, Ueber den Verband der orthogonal abgeschlossenen Teilräume eines hermiteschen Raumes. Letter to the author of Nov. 7 1973 pp. 1–6.

    Google Scholar 

  26. G. Käthe, Topological Vector Spaces I. Grundlehren Band 159, Springer Verlag, Heidelberg New York 1969.

    Google Scholar 

  27. S. Lefschetz, Algebraic Topology. AMS Colloquium Publ. vol.XXVII. Reprinted 1963 by AMS New York.

    Google Scholar 

  28. E.A. Lüssi, Ueber Cliffordalgebren als quadratische Räume. Ph.D. Thesis, University of Zurich 1971.

    Google Scholar 

  29. M.D. Mac Laren, Atomic orthocomplemented lattices. Pacific J. Math. 14 (1964) 597–612.

    Google Scholar 

  30. F. Maeda and S. Maeda, Theory of symmetric lattices. Grundlehren Band 173, Springer, Berlin Heidelberg New York 1970.

    Google Scholar 

  31. G. Maxwell, Infinite symplectic groups over rings. Comment. Math. Helv. 47 (1972) 254–259.

    Google Scholar 

  32. E. Ogg, Ein Satz über orthogonal abgeschlossene Unterräume. Comment. Math. Helv. 44 (1969) 117–119.

    Article  MATH  Google Scholar 

  33. O.T. O’Meara, Symplectic groups. Math. Surveys vol. 16, AMS Providence R. I. 1978.

    Google Scholar 

  34. V. Pless, On Witt’s theorem for nonalternating symmetric bilinear forms over a field of characteristic 2. Proc. Amer. Math. Soc. 15 (1964) 979–983.

    MathSciNet  MATH  Google Scholar 

  35. V. Pless, On the invariants of a vector subspace of a vector space over a field of characteristic two. Proc. Amer. Math. Soc. 16 (1965) 1062–1067.

    Article  MathSciNet  MATH  Google Scholar 

  36. C.E. Rickart, Isomorphisms of infinite–dimensional analogues of the classical groups. Bull. Amer. Math. Soc. 57 (1951) 435–448.

    Google Scholar 

  37. J. Saranen, Ueber die Verbandcharakterisierung einiger nichtentarteter Formen. Ann. Acad. Sci. Fenn. Ser. A.I vol. 1 (1975) 85–92.

    MATH  Google Scholar 

  38. W. Scharlau, Zur Existenz von Involutionen auf einfachen Algebren. Math. Z. 145 (1975) 29–32.

    Article  MathSciNet  MATH  Google Scholar 

  39. J.A. Schouten, Ricci-Calculus. 2nd ed. Grundlehren vol. 10, Springer, Berlin Heidelberg 1954.

    Google Scholar 

  40. U. Schneider, Ueber Räume mit wenig orthogonalen Zerlegungen. Masters Thesis, University of Zurich 1972.

    Google Scholar 

  41. E. Witt, Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937) 31–44.

    Google Scholar 

  42. W. Meissner, Untersuchungen unendlich dimensionaler quadratischer Räume im Hinblick auf modelltheoretische Uebertragungsprinzipien. This Ph.D. thesis (University of Konstanz) is nearing completion. Refer to forthcoming publications by Meissner.

    Google Scholar 

References to Appendix I

  1. R. Baer, Linear Algebra and Projective Geometry. Academic Press Inc., New York 1952.

    MATH  Google Scholar 

  2. L.E. Dickson, Algebren und ihre Zahlentheorie. Orell Füssli Verlag Zürich (Switzerland ) 1927.

    Google Scholar 

  3. H. Gross and H.A. Keller, On the definition of Hilbert space. Manuscripta math. 23 (1977) 67–90.

    Article  MathSciNet  MATH  Google Scholar 

  4. S.S. Holland, Orderings and Square Roots in *-Fields. J. Algebra 46 (1977) 207–219.

    Article  MathSciNet  MATH  Google Scholar 

  5. S.S. Holland, Orthomodular forms over ordered *-fields. To appear.

    Google Scholar 

  6. A. Prestel, Lectures on Formally real Fields. Monografias de Mat. 22, Inst. de Mat. Pura e Aplicada, Rio de Janeiro 1975.

    Google Scholar 

  7. A. Prestel, Quadratische Semiordnungen und quadratische Formen. Math. Z. 133 (1973) 319–342.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Prestel, Euklidische Geometrie ohne das Axiom von Pasch. Abh. Math. Sem. Hamburg 41 (1974) 82–109.

    Article  MATH  Google Scholar 

  9. W.J. Wilbur, On characterizing the standard quantum logics. Trans. Amer. Math. Soc. 233 (1977) 265–282.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Zürich, Freiestrasse 36, CH-8032, Zürich, Switzerland

    Herbert Gross

Authors
  1. Herbert Gross
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1979 Springer Science+Business Media New York

About this chapter

Cite this chapter

Gross, H. (1979). Fundamentals on Sesquilinear Forms. In: Quadratic Forms in Infinite Dimensional Vector Spaces. Progress in Mathematics, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1454-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-1454-8_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-1456-2

  • Online ISBN: 978-1-4757-1454-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation