An Inner Product on the Complex Numbers in Rn

We cannot define a dot product o
$\mathbb{C}^n$
, the set
$\{ ( c_1,...,c_n ) \colon c_i \in \mathbb{C} , i =1,2,...,n \}$
to satisfy the same set of inner product requirements as for the vectors in
$\mathbb{R}^n$
, but can can come close. We can define an inner product on
$\mathbb{C}^n$
to satisfy
$\langle \mathbf{u} , \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle^* =\mathbf{u} \cdot \mathbf{v}^*$

Symmetry: If
$c\mathbf{u}, \: \mathbf{v} \in \mathbb{C}^n$
then
$\langle \mathbf{u} , \mathbf{v} \rangle = \langle \mathbf{v} , \mathbf{u} \rangle$

Positive definite: If
$c\mathbf{u} \in \mathbb{C}^n$
then
$\langle \mathbf{u} , \mathbf{u} \rangle = \mathbf{u} \cdot \mathbf{u}^* \geq 0$

Since
$\mathbf{u} \cdot \mathbf{u} =0$
if and only if
$\mathbf{u} = \mathbf{u}^0 = \mathbf{0}$

Linear in both arguments: Let
$\mathbf{u} = (u_1, ...,u_n), v =(v_1,...,v_n)$
then
$\langle \mathbf{u} , \mathbf{v} \rangle = \sum_i^n u_i v^*_i$

$\alpha \sum_i^n u_i v^*_i = \langle \alpha \mathbf{u} , \mathbf{v} \rangle = \sum_i^n \alpha u_i v^*_i = \sum_i^n u_i \alpha v^*_i$