(i) Let $H$ be an infinite-dimensional Hilbert space. Show that $H$ has a (countable) orthonormal basis if and only if $H$ has a countable dense subset. [You may assume familiarity with the Gram-Schmidt process.]

State and prove Bessel's inequality.

(ii) State Parseval's equation. Using this, prove that if $H$ has a countable dense subset then there is a surjective isometry from $H$ to $l^{2}$.

Explain carefully why the functions $e^{i n \theta}, n \in \mathbb{Z}$, form an orthonormal basis for $L^{2}(\mathbb{T})$