(a) Let $\left(e_{n}\right)_{n=1}^{\infty}$ be an orthonormal basis of an inner product space $X$. Show that for all $x \in X$ there is a unique sequence $\left(a_{n}\right)_{n=1}^{\infty}$ of scalars such that $x=\sum_{n=1}^{\infty} a_{n} e_{n}$.

Assume now that $X$ is a Hilbert space and that $\left(f_{n}\right)_{n=1}^{\infty}$ is another orthonormal basis of $X$. Prove that there is a unique bounded linear map $U: X \rightarrow X$ such that $U\left(e_{n}\right)=f_{n}$ for all $n \in \mathbb{N}$. Prove that this map $U$ is unitary.

(b) Let $1 \leqslant p<\infty$ with $p \neq 2$. Show that no subspace of $\ell_{2}$ is isomorphic to $\ell_{p}$. [Hint: Apply the generalized parallelogram law to suitable vectors.]