Axiom:Complex Semi-Inner Product Axioms

Definition

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.

Definition:Complex Inner Product, a semi-inner product with the additional property of positiveness.

\((1)\)	$:$	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	\(\ds \forall x \in V:\)	\(\ds \quad \innerprod x x \in \R_{\ge 0} \)