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.
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References to Chapter I
A.A. Albert, Structure of Algebras. (3rd print of rev.ed.) AMS Coll Publ. XXIV, New York 1968.
R. Baer, Linear Algebra and Projective Geometry. Academic Press, New York 1952.
W. Baur and H. Gross, Strange inner product spaces. Comment. Math. Heiv. 52 (1977) 491–495.
G. Birkhoff, Lattice Theory. AMS Coll Publ. XXV (3rd ed,2nd print) Providence, R. I. 1973.
G. Birkhoff, and J. v. Neumann, The logic of quantum mechanics. Ann. of Math. 37 (1936) 823–843.
N. Bourbaki, Formes sesquilinéaires et formes quadratiques. Hermann Paris 1959.
J. Dieudonné, On the structure of unitary groups. Trans. Amer. Math. Soc. 72 (1952) 367–385.
J. Dieudonné, On the structure of unitary groups II. Amer. J. Math. 75 (1953) 665–678.
J. Dieudonné, La géometrie des groupes classiques, 3ième éd. Ergebnisse der Mathematik, Heft 5, Springer Berlin, Heidelberg 1971.
H.R. Fischer and H. Gross, Quadratic Forms and Linear Topologies I. Math. Ann. 157 (1964) 296–325.
H.R. Fischer and H. Gross, Tensorprodukte linearer Topologien (Quadratische Formen und lineare Topologien III). Math. Ann. 160 (1965) 1–40.
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)
H. Gross, On a representation theorem for AC-lattices. Mimeographed notes 1974.
H. Gross, Linearly topologized spaces without continuous bases (Quadratic forms and linear topologies V). Math. Ann. 194 (1971) 313–315.
H. Gross, and H.A. Keller, On the definition of Hilbert space. Manuscripta math. 23 (1977) 67–90.
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.
H. Gross, and E. Ogg, On completions. Ann. Acad. Sci. Fenn. Ser. A. I 584 (1974) 1–19.
H. Gross, and E. Ogg,, Quadratic spaces with few isometries. Comment. Math. Hell/. 48 (1973) 511–519.
O. Hamara, On the structure of the orthogonal group. Math. Scand. 21 (1967) 219–232.
O. Hamara, Quadratic forms on linearly topologized vector spaces. Portugal. Math. 27 (1968) 15–30.
I.N. Herstein, On a Theorem of Albert. Scripta Mathematica XXIX (1973) 391–394.
I. Kaplansky, Forms in infinite dimensional spaces. Anais da Academia Brasileira de Ciencias 22 (1950) 1–17.
I. Kaplansky,, Linear Algebra and Geometry. Allyn and Bacon, Boston 1969.
H.A. Keller, Stetigkeitsfragen bei lineartopologischen Cliffordalgebren. Ph.D. Thesis, University of Zurich 1971.
H.A. Keller, Ueber den Verband der orthogonal abgeschlossenen Teilräume eines hermiteschen Raumes. Letter to the author of Nov. 7 1973 pp. 1–6.
G. Käthe, Topological Vector Spaces I. Grundlehren Band 159, Springer Verlag, Heidelberg New York 1969.
S. Lefschetz, Algebraic Topology. AMS Colloquium Publ. vol.XXVII. Reprinted 1963 by AMS New York.
E.A. Lüssi, Ueber Cliffordalgebren als quadratische Räume. Ph.D. Thesis, University of Zurich 1971.
M.D. Mac Laren, Atomic orthocomplemented lattices. Pacific J. Math. 14 (1964) 597–612.
F. Maeda and S. Maeda, Theory of symmetric lattices. Grundlehren Band 173, Springer, Berlin Heidelberg New York 1970.
G. Maxwell, Infinite symplectic groups over rings. Comment. Math. Helv. 47 (1972) 254–259.
E. Ogg, Ein Satz über orthogonal abgeschlossene Unterräume. Comment. Math. Helv. 44 (1969) 117–119.
O.T. O’Meara, Symplectic groups. Math. Surveys vol. 16, AMS Providence R. I. 1978.
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.
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.
C.E. Rickart, Isomorphisms of infinite–dimensional analogues of the classical groups. Bull. Amer. Math. Soc. 57 (1951) 435–448.
J. Saranen, Ueber die Verbandcharakterisierung einiger nichtentarteter Formen. Ann. Acad. Sci. Fenn. Ser. A.I vol. 1 (1975) 85–92.
W. Scharlau, Zur Existenz von Involutionen auf einfachen Algebren. Math. Z. 145 (1975) 29–32.
J.A. Schouten, Ricci-Calculus. 2nd ed. Grundlehren vol. 10, Springer, Berlin Heidelberg 1954.
U. Schneider, Ueber Räume mit wenig orthogonalen Zerlegungen. Masters Thesis, University of Zurich 1972.
E. Witt, Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937) 31–44.
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.
References to Appendix I
R. Baer, Linear Algebra and Projective Geometry. Academic Press Inc., New York 1952.
L.E. Dickson, Algebren und ihre Zahlentheorie. Orell Füssli Verlag Zürich (Switzerland ) 1927.
H. Gross and H.A. Keller, On the definition of Hilbert space. Manuscripta math. 23 (1977) 67–90.
S.S. Holland, Orderings and Square Roots in *-Fields. J. Algebra 46 (1977) 207–219.
S.S. Holland, Orthomodular forms over ordered *-fields. To appear.
A. Prestel, Lectures on Formally real Fields. Monografias de Mat. 22, Inst. de Mat. Pura e Aplicada, Rio de Janeiro 1975.
A. Prestel, Quadratische Semiordnungen und quadratische Formen. Math. Z. 133 (1973) 319–342.
A. Prestel, Euklidische Geometrie ohne das Axiom von Pasch. Abh. Math. Sem. Hamburg 41 (1974) 82–109.
W.J. Wilbur, On characterizing the standard quantum logics. Trans. Amer. Math. Soc. 233 (1977) 265–282.
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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
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