$\langle x,y \rangle + \langle y,x \rangle = 2\operatorname{Re}\langle x,y \rangle$

Question

In an inner product space,

$\langle x,y \rangle + \langle y,x \rangle = 2\operatorname{Re}\langle x,y \rangle$

and

$\langle x,-y \rangle + \langle -y,x \rangle = -2\operatorname{Re}\langle y,x \rangle$

Why are these two identities true? What is the significance of writing Re instead of leaving the sum as $\langle x,y \rangle + \langle y,x \rangle$?

Answer 1

Inner product over $C$ is not symmetric but hermitian, meaning that $⟨y,x⟩ = \overline{⟨x,y⟩}$ So $⟨x,y⟩+⟨y,x⟩ = ⟨x,y⟩ + \overline{⟨x,y⟩} = 2*Re(⟨x,y⟩)$. Writing Re means that you have a real number on the right hand side where as on the left hand side you might have complex numbers.

Hope this helps.