I would like to show that for any sequence $f_n(x) \in L^2([0,1])$ with $||f_n||_2=1$ for all $n$, there exists an $f\in L^2([0,1])$ and a subsequence $(f_{n_k})$ such that for every $g\in L^2([0,1])$ one has:
$lim_{k \rightarrow \infty} (f_{n_k}, g) = (f,g)$
Here $(f,g) = \int_0^1 f(x)\cdot g(x) dx$ is the scalar product.
The given solution states: Let $(g_j)$ enumerate a countable dense subset. Then $(f_n,g_1)$ is totally bounded and so has a convergent subsequence. It then goes on to say that from that subsequence $(f_{n_k},g_2)$ is also totally bounded at which point I no longer follower their proof.
Any explanation or a different proof would be greatly appreciated!