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

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