Abstract
Let E be a vector space over the division ring k and L(E) the lattice of all linear subspaces of E. If Ē is a vector space over the division ring k̄ and τ: L(E) → L(Ē) a lattice isomorphism then by the Fundamental Theorem of Projective Geometry ([1] p. 44) τ is induced by a semilinear map T: E → Ē if we assume that dim E ≥ 3.
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References to Chapter IV
R. Baer, Linear Algebra and Projective Geometry. Academic Press, New York 1952.
H. Gross, On Witt’s Theorem in the Denumerably Infinite Case. Math. Ann. 170 (1967) 145–165.
H. Gross, Isomorphisms between lattices of linear subspaces which are induced by Isometries. J. Algebra 49 (1977) 537–546.
H. Gross and H.A. Keller, On the definition of Hilbert Space. manuscripta math. 23 (1977) 67–90.
C. Herrmann, On a condition sufficient for the distributivity of lattices of linear subspaces. To appear.
P. Pudlak and J. Tůma, Yeast graphs and fermentation of algebraic lattices. Coll. Math. Soc. J. Bolyai, 14 (1976) Lattice Theory 301–342 ed. by A.P. Huhn and E.T. Schmidt, North Holland Publ. Company, Amsterdam.
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© 1979 Springer Science+Business Media New York
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Gross, H. (1979). Isomorphisms between Lattices of Linear Subspaces Which are Induced by Isometries. In: Quadratic Forms in Infinite Dimensional Vector Spaces. Progress in Mathematics, vol 1. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3542-7_5
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DOI: https://doi.org/10.1007/978-1-4899-3542-7_5
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-1111-8
Online ISBN: 978-1-4899-3542-7
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