Exercise 17, § 3

Let $n\geqslant 1$ be an integer. Let $\mu$ denote the Lebesgue measure on $\mathbf{R}^n$.

a) A bounded subset B of $L^2(\mathbf{R}^n, \mu)$ is compact if, and only if, for every $\varepsilon >0$, there exists a compact subset K of $\mathbf{R}^n$ such that

$\mathbf{R}^n-$K $\mathbf{R}^n-$K

for every $f\in B$.

b) Let $k\in \mathbf{N}$. Let K be a compact subset of $\mathbf{R}^n$. Denote by $H^2_k(K)$ the closed subspace of the Sobolev space $H^2_k(\mathbf{R}^n)$ consisting of functions vanishing outside the compact subset K. The canonical injection $H^2_k(K)\rightarrow H^2_k(\mathbf{R}^n)$ is compact.