Classification of Subspaces in Spaces with Definite Forms

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


In the whole chapter (E, Φ) will be a positive definite hermitean space of dimension אo over the divisionring k with involution ξ ⟼ ξτ. If τ ≠ 1 then it follows from Dieudonné’s lemma that k is either a quadratic extension k = ko (γ) over an ordered field (ko, <) with 0 > γ2 ∈ ko and (x+yγ)τ = x-yγ for all x, y ∈ ko: or k is a quaternion algebra \((\frac{{\alpha ,\beta }} {{{k_0}}})\) with ko ordered, α, β < 0 and τ being the usual “conjugation”. If τ = 1, possible only when k is commutative, then ϕ is symmetric and k = ko is ordered.


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Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • Herbert Gross
    • 1
  1. 1.Mathematisches InstitutUniversität ZürichZürichSwitzerland

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