Density of Sobolev spaces

Question

Is the Sobolev space $H^s(\mathbb{R}^3)$ dense in $L^2(\mathbb{R}^3)$ for $s>0$ ?

Answer 1

We denote by $\mathcal D(\Bbb R^d)$ the set of smooth functions with compact support.

• As the Fourier transform of a function in $\mathcal D(\Bbb R^d)$ has an arbitrary decay, we have the inclusion $\mathcal D(\Bbb R^d)\subset H^s(\Bbb R^d)$ for any $s>0$.
• By Plancherel's formula, we have $H^s(\Bbb R^d)\subset L^2(\Bbb R^d)$.
• By a truncation and regularization argument, it can be shown that $\mathcal D(\Bbb R^d)$ in dense in $L^2(\Bbb R^d)$ with its natural norm.