in order for me to understand this theorim

do i have to think of the vectors as complex??

what the values of x1 ,y1 ,x2 ,y2

$\displaystyle

\langle x , y \rangle = x_1^* \cdot y_1 + x_2^*\cdot y_2 + \ldots

$

did i get the structure correctly

x=(x1,y1) y=(y1,y2) each one of x1,x2,y1,y2 is a complex number??