$L^2$ functions with compactly supported Fourier transforms form a Hilbert space

Question

Given a fixed compact subset of $\mathbb{R}$, I want to show that square integrable functions on the real line whose fourier transforms are supported in the given compact set form a Hilbert space in the $L^2$ inner product. It was easy to show that the functions form a subspace. However I am stuck at the closedness of the subspace. Could anyone please help me?

Answer 1

$f_n \to f$ in $L^{2}$ implies $\hat {f_n} \to \hat {f}$ in $L^{2}$ and this implies that some subequence of $(\hat {f_n})$ converges almost everywhere to $\hat {f}$. Hence $\hat {f}$ is also supported by the compact set.