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Linear Algebra pp 325-350 | Cite as

Unitary spaces

  • Werner Greub
Part of the Graduate Texts in Mathematics book series (GTM, volume 23)

Abstract

Let E be an n-dimensional complex linear space and Φ: E × E→ ℂ be a function such that
$$\Phi \left( {\lambda {{\text{x}}_{\text{1}}} + \mu {{\text{x}}_{\text{2}}},{\text{y}}} \right) = {\text{ }}\lambda {\text{ }}\Phi \left( {{{\text{x}}_{\text{1}}},{\text{y}}} \right){\text{ }} + \mu {\text{ }}\Phi \left( {{{\text{x}}_{\text{2}}},{\text{y}}} {x,\lambda {y_{\text{1}}} + \mu {{\text{y}}_2}} \right) = {\text{ }}\bar \lambda {\text{ }}\Phi \left( {{\text{x}},{{\text{y}}_2}} \right){\text{ }} + \bar \mu {\text{ }}\Phi \left( {{\text{x}},{{\text{y}}_2}} \right)$$
(11.1)
where λ̄ and μ̄ are the complex conjugate coefficients. Then Φ will be called a sesquilinear function. Replacing y by x we obtain from Φ the corresponding quadratic function
$$\Psi \left( {\text{x}} \right) = {\text{ }}\Phi \left( {{\text{x}},{\text{x}}} \right)$$
(11.2)
.

Keywords

Orthonormal Basis Linear Transformation Unitary Mapping Real Form Real Vector Space 
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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Copyright information

© Springer-Verlag New York Inc. 1975

Authors and Affiliations

  • Werner Greub
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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