If $B$ is an orthonormal basis

Question

Let $V$ be a unitary space and $x,y \in V$. Prove that if $\{b_1, b_2,...,b_n\}$ is an orthonormal basis for $V$ then $\langle x,y \rangle = \sum\limits_{i=1}^{n}\langle x,b_i \rangle \overline{\langle y,b_i \rangle}$.

Answer 1

Hint:

Since $\{ b_1,\ldots,b_n\}$ is an orthonormal basis, you can write

$$x = \sum x_i b_i$$ $$y = \sum y_i b_i$$

for unique choices of coefficients $x_i,y_i$ with $1\leq i \leq n$. Now prove that $x_i = \langle x,b_i \rangle$ and $y_i = \langle y,b_i \rangle$ and then express $\langle x,y \rangle$ using the above linear combinations.