Unitary spaces

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))

.