Let $\mathbb{C}$ be the D652: Ring of complex numbers such that

(i)	$V$ is a D29: Vector space over $\mathbb{C}$
(ii)	$0_V$ is an D270: Additive identity in $V$

A D4881: Complex function $f : V \times V \to \mathbb{C}$ is an inner product on $V$ over $\mathbb{C}$ if and only if

(1)	$\forall \, x, y, z \in V : f(x + y, z) = f(x, z) + f(y, z)$	(D678: Additive map)
(2)	$\forall \, x, y, z \in V : f(x, y + z) = f(x, y) + f(x, z)$	(D678: Additive map)
(3)	$\forall \, x, y \in V : \forall \, r \in \mathbb{C} : f(r x, y) = r f(x, y)$	(D983: Homogeneous map)
(4)	$\forall \, x, y \in V : \forall \, r \in \mathbb{C} : f(x, r y) = \overline{r} f(x, y)$	(D987: Conjugate homogeneous map)
(5)	$\forall \, x, y \in V : f(x, y) = \overline{f(y, x)}$	(D729: Conjugate symmetric complex function)
(6)	$\forall \, x \in V : f(x, x) \geq 0$
(7)	$\forall \, x \in V \left( f(x, x) = 0 \quad \implies \quad x = 0_V \right)$