\[\mathbb{C}^n\]
, 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 \[\langle \mathbf{u} , \mathbf{v} \rangle = \sum_i^n u_i v^*_i\]
\[\alpha \sum_i^n u_i v^*_i = \langle \alpha \mathbf{u} , \mathbf{v} \rangle = \sum_i^n \alpha u_i v^*_i = \sum_i^n u_i \alpha v^*_i\]