Let $\mathbb{C}$ be the
D652: Ring of complex numbers such that
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) |
$\forall \, x \in V
\left( f(x, x) = 0 \quad \implies \quad x = 0_V \right)$
|
|
