Exercise 5.2.3

Let $v_1,v_2,...,v_n$ be an orthonormal basis in $V$.

a)

Prove that for any $x={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{k=1}^n{\alpha }_k{v}_k$, $y={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{k=1}^n{\beta }_k{v}_k$

$$(x,y)=\sum _{k=1}^n{\alpha }_k{\overline{\beta }}_k.$$

b)

Deduce from this Parseval’s identity

$$(x,y)=\sum _{k=1}^n({x,v}_k)\overline{({y,v}_k)}.$$

c)

Assume now that $v_1,v_2,...,v_n$ is only an orthogonal basis, not an orthonormal one. Can you write down Parseval’s identity in this case?