Set of fuctions that are differentiable and have their derivative in $L^2(\mathbb R)$ is a dense set in $L^2(\mathbb R)$

Question

Let $S=\{x\in L^2(\mathbb R):\frac{dx}{dt} \text{ exists},\frac{dx}{dt}\in L^2(\mathbb R)\}$.

I want to prove that $S$ is dense in $L^2(\mathbb R)$. The norm is the usual $L^2$ norm: $$ \|x\|_2=\left(\int_\mathbb R |x(t)|^2dt\right)^{1/2}. $$

I know how to prove this for $L^2[a,b]$ by approximating with polynomials with Stone–Weierstrass theorem. A weaker result can be found here. But I struggle to play with polynomials in $\mathbb R$, since no non-zero polynomials are in $L^2(\mathbb R)$.

How to prove that $S$ is dense in $L^2(\mathbb R)$? Or can anyone prove a stronger result? It seems there are proper dense subsets of $S$.

Answer 1

One can show that $C_c^\infty(\mathbb R)$, the space of infinitely differentiable, compactly supported functions on $\mathbb R$, is dense in $L^2(\mathbb R)$. Take any $f\in L^2(\mathbb R)$, approximate it in the $L^2$-norm by a simple function, and approximate the simple function by something in $C_c^\infty(\mathbb R)$, using regularity properties of Lebesgue measure and a $C^\infty$ version of the Urysohn lemma.