Let ${f_n}$ a bounded sequence is $L^2(\mathbb{R})$ such that $\int f_n \phi dx \rightarrow0$ for each $\phi \in C_c^\infty(\mathbb{R})$. I want to prove that $f_n \rightharpoonup 0$.
The sequence is bounded so exists $f \in L^2$ and a subsequence $f_{n_k}$ of $f_n$ sucht that $f_{n_k} \rightharpoonup f$. If we pick $\phi \in C^\infty_c$ then $\int f_{n_k} \phi dx \rightarrow \int f \phi dx$ (and $\int f_{n_k} \phi dx \rightarrow 0$) so $\int f \phi dx=0$ for each $\phi \in C^\infty_c$ which implies $f=0$ a.e.
This proves that every convergent subsequence of $f_n$ converge to $0$. How can I conclude?
Best Answer
Let $g \in L^{2}$ and $\epsilon >0$. There exists $\phi \in C_c^{\infty} (\mathbb R)$ such that $\|g-\phi\|_2 <\epsilon$. Hence $|\int f_n g|\leq |\int f_n \phi|+\|f_n\|\epsilon$ (by Hölder's/ C-S inequality). It is clear now that $\int f_n g \to 0$.