Let B = {v 1 ,......, v n } be an orthonormal basis for R n .

Question:

Let B = {v₁,......, v_n} be an orthonormal basis for Rⁿ.

(a) Prove that, for any x and y in Rⁿ, x • y = (x • v₁) (y • v₁) + (x • v₂) (y • v₂) + ...........+ (x • v_n) (y • v_n) (This identity is called Parseval’s Identity.)
(b) What does Parseval’s Identity imply about the relationship between the dot products x • y and [x]_B • [y]_B?