Axiom:Complex Semi-Inner Product Axioms

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Definition

Let $V$ be a vector space over a complex subfield $\GF$.

Let $\innerprod \cdot \cdot: V \times V \to \GF$ be a mapping.


The mapping $\innerprod \cdot \cdot$ is a complex semi-inner product if and only if $\innerprod \cdot \cdot$ satisfies the following axioms:

\((1)\)   $:$   Conjugate Symmetry      \(\ds \forall x, y \in V:\) \(\ds \quad \innerprod x y = \overline {\innerprod y x} \)      
\((2)\)   $:$   Sesquilinearity      \(\ds \forall x, y, z \in V, \forall a \in \GF:\) \(\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z \)      
\((3)\)   $:$   Non-Negative Definiteness      \(\ds \forall x \in V:\) \(\ds \quad \innerprod x x \in \R_{\ge 0} \)      

These criteria are called the (complex) semi-inner product axioms.


Also see