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Fundamentals on Sesquilinear Forms

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

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.

Keywords

Linear Subspace Division Ring Hyperbolic Plane Hermitean Form Finite Dimensional Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References to Chapter I

  1. [1]
    A.A. Albert, Structure of Algebras. (3rd print of rev.ed.) AMS Coll Publ. XXIV, New York 1968.Google Scholar
  2. [2]
    R. Baer, Linear Algebra and Projective Geometry. Academic Press, New York 1952.zbMATHGoogle Scholar
  3. [3]
    W. Baur and H. Gross, Strange inner product spaces. Comment. Math. Heiv. 52 (1977) 491–495.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    G. Birkhoff, Lattice Theory. AMS Coll Publ. XXV (3rd ed,2nd print) Providence, R. I. 1973.Google Scholar
  5. [5]
    G. Birkhoff, and J. v. Neumann, The logic of quantum mechanics. Ann. of Math. 37 (1936) 823–843.MathSciNetCrossRefGoogle Scholar
  6. [6]
    N. Bourbaki, Formes sesquilinéaires et formes quadratiques. Hermann Paris 1959.Google Scholar
  7. [7]
    J. Dieudonné, On the structure of unitary groups. Trans. Amer. Math. Soc. 72 (1952) 367–385.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. Dieudonné, On the structure of unitary groups II. Amer. J. Math. 75 (1953) 665–678.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Dieudonné, La géometrie des groupes classiques, 3ième éd. Ergebnisse der Mathematik, Heft 5, Springer Berlin, Heidelberg 1971.Google Scholar
  10. [10]
    H.R. Fischer and H. Gross, Quadratic Forms and Linear Topologies I. Math. Ann. 157 (1964) 296–325.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    H.R. Fischer and H. Gross, Tensorprodukte linearer Topologien (Quadratische Formen und lineare Topologien III). Math. Ann. 160 (1965) 1–40.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [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. [13]
    H. Gross, On a representation theorem for AC-lattices. Mimeographed notes 1974.Google Scholar
  14. [14]
    H. Gross, Linearly topologized spaces without continuous bases (Quadratic forms and linear topologies V). Math. Ann. 194 (1971) 313–315.Google Scholar
  15. [15]
    H. Gross, and H.A. Keller, On the definition of Hilbert space. Manuscripta math. 23 (1977) 67–90.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    H. Gross, and E. Ogg, On completions. Ann. Acad. Sci. Fenn. Ser. A. I 584 (1974) 1–19.MathSciNetGoogle Scholar
  18. [18]
    H. Gross, and E. Ogg,, Quadratic spaces with few isometries. Comment. Math. Hell/. 48 (1973) 511–519.CrossRefzbMATHGoogle Scholar
  19. [19]
    O. Hamara, On the structure of the orthogonal group. Math. Scand. 21 (1967) 219–232.MathSciNetzbMATHGoogle Scholar
  20. [20]
    O. Hamara, Quadratic forms on linearly topologized vector spaces. Portugal. Math. 27 (1968) 15–30.MathSciNetzbMATHGoogle Scholar
  21. [21]
    I.N. Herstein, On a Theorem of Albert. Scripta Mathematica XXIX (1973) 391–394.Google Scholar
  22. [22]
    I. Kaplansky, Forms in infinite dimensional spaces. Anais da Academia Brasileira de Ciencias 22 (1950) 1–17.MathSciNetGoogle Scholar
  23. [23]
    I. Kaplansky,, Linear Algebra and Geometry. Allyn and Bacon, Boston 1969.zbMATHGoogle Scholar
  24. [24]
    H.A. Keller, Stetigkeitsfragen bei lineartopologischen Cliffordalgebren. Ph.D. Thesis, University of Zurich 1971.Google Scholar
  25. [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. [26]
    G. Käthe, Topological Vector Spaces I. Grundlehren Band 159, Springer Verlag, Heidelberg New York 1969.Google Scholar
  27. [27]
    S. Lefschetz, Algebraic Topology. AMS Colloquium Publ. vol.XXVII. Reprinted 1963 by AMS New York.Google Scholar
  28. [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. [30]
    F. Maeda and S. Maeda, Theory of symmetric lattices. Grundlehren Band 173, Springer, Berlin Heidelberg New York 1970.Google Scholar
  31. [31]
    G. Maxwell, Infinite symplectic groups over rings. Comment. Math. Helv. 47 (1972) 254–259.Google Scholar
  32. [32]
    E. Ogg, Ein Satz über orthogonal abgeschlossene Unterräume. Comment. Math. Helv. 44 (1969) 117–119.CrossRefzbMATHGoogle Scholar
  33. [33]
    O.T. O’Meara, Symplectic groups. Math. Surveys vol. 16, AMS Providence R. I. 1978.Google Scholar
  34. [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.MathSciNetzbMATHGoogle Scholar
  35. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    C.E. Rickart, Isomorphisms of infinite–dimensional analogues of the classical groups. Bull. Amer. Math. Soc. 57 (1951) 435–448.Google Scholar
  37. [37]
    J. Saranen, Ueber die Verbandcharakterisierung einiger nichtentarteter Formen. Ann. Acad. Sci. Fenn. Ser. A.I vol. 1 (1975) 85–92.zbMATHGoogle Scholar
  38. [38]
    W. Scharlau, Zur Existenz von Involutionen auf einfachen Algebren. Math. Z. 145 (1975) 29–32.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    J.A. Schouten, Ricci-Calculus. 2nd ed. Grundlehren vol. 10, Springer, Berlin Heidelberg 1954.Google Scholar
  40. [40]
    U. Schneider, Ueber Räume mit wenig orthogonalen Zerlegungen. Masters Thesis, University of Zurich 1972.Google Scholar
  41. [41]
    E. Witt, Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937) 31–44.Google Scholar
  42. [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. [1]
    R. Baer, Linear Algebra and Projective Geometry. Academic Press Inc., New York 1952.zbMATHGoogle Scholar
  2. [2]
    L.E. Dickson, Algebren und ihre Zahlentheorie. Orell Füssli Verlag Zürich (Switzerland ) 1927.Google Scholar
  3. [3]
    H. Gross and H.A. Keller, On the definition of Hilbert space. Manuscripta math. 23 (1977) 67–90.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S.S. Holland, Orderings and Square Roots in *-Fields. J. Algebra 46 (1977) 207–219.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S.S. Holland, Orthomodular forms over ordered *-fields. To appear.Google Scholar
  6. [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. [7]
    A. Prestel, Quadratische Semiordnungen und quadratische Formen. Math. Z. 133 (1973) 319–342.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Prestel, Euklidische Geometrie ohne das Axiom von Pasch. Abh. Math. Sem. Hamburg 41 (1974) 82–109.CrossRefzbMATHGoogle Scholar
  9. [9]
    W.J. Wilbur, On characterizing the standard quantum logics. Trans. Amer. Math. Soc. 233 (1977) 265–282.MathSciNetCrossRefzbMATHGoogle 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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