Axiom:Real Inner Product Axioms

Definition

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

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

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

$(1')$	$:$	Symmetry	$\ds \forall x, y \in V:$	$\ds \innerprod x y = \innerprod y x $
$(2)$	$:$	Linearity in first argument	$\ds \forall x, y \in V, \forall a \in \GF:$	$\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} $
$(4)$	$:$	Positiveness	$\ds \forall x \in V:$	$\ds \quad \innerprod x x = 0 \implies x = \mathbf 0_V $

These criteria are called the (real) inner product axioms.

\((1')\)	$:$	Symmetry	\(\ds \forall x, y \in V:\)	\(\ds \innerprod x y = \innerprod y x \)
\((2)\)	$:$	Linearity in first argument	\(\ds \forall x, y \in V, \forall a \in \GF:\)	\(\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z \)
\((3)\)	$:$	Non-Negative Definiteness	\(\ds \forall x \in V:\)	\(\ds \quad \innerprod x x \in \R_{\ge 0} \)
\((4)\)	$:$	Positiveness	\(\ds \forall x \in V:\)	\(\ds \quad \innerprod x x = 0 \implies x = \mathbf 0_V \)