$f_n \in C_0^\infty(0,1)$, and $M>0$ such that $\int_0^1|f_n(x)|^2dx+\int_0^1|f_n'(x)|^2dx<M^2$, $\forall n$. Prove that $\{f_n\}$ has a subsequence $\{f_{n_k}\}$ such that $\forall g\in L^2(0,1)$, $$\lim_{k \to \infty}\langle g,f_{n_k}\rangle =\langle g,f\rangle,$$ where $\langle f,g\rangle=\int_0^1f(x)\overline {g(x)} dx$.
The previous question is to prove that $\forall x \in (0,1), |f_n(x)|\le M (n=1,2,\cdots)$, which I have already proved. And the questions after this question are to prove that $\{f_{n_k}\}$ is a Cauchy sequence, and therefore $\{f_{n_k}\}$ converges in $L^2(0,1)$.
For this question, I try to use that $e^{inx} $ is an orthogonal basis of $L^2[0,2\pi]$, so there exists $\{\varphi_n\}$ the orthonormal basis of $L^2(0,1)$. (Is this true?)
Then I can take a subsequence $\{f_{n_k}\}$ such that $\lim_{k\to \infty}\langle \varphi_m,f_{n_k} \rangle$ converges for each $\varphi_m$. Let $\lim_{k\to \infty}\langle \varphi_m,f_{n_k} \rangle=\alpha_m$. (using the diagonal method)
Then take $f=\sum_{m=1}^\infty \alpha_m\varphi_m$, then $\forall m,\lim_{k\to \infty}\langle \varphi_m,f_{n_k} \rangle=\langle \varphi_m,f\rangle$, and therefore $\forall g,\lim_{k\to \infty}\langle g,f_{n_k} \rangle=\langle g,f\rangle$. But $\sum_{m=1}^\infty \alpha_m\varphi_m$ may not converge? So I don't know whether this method works.
How to solve this question? Many thanks in advance! (By the way, I just begin learning Hilbert space.)
