Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 23F / a / Solution

Choose a nonzero $\varphi \in C_c^{\infty }({\mathbb{R}}^n)$ supported in a ball of radius $1$, and letTranslation preserves both the $L^2$ norm and the weak-derivative norms, sofor every $k$. Thus $({\varphi }_k)$ is bounded in the Sobolev space $H^1({\mathbb{R}}^n)$.

Distinct translates have disjoint supports. Hence, for $j\ne k$,No subsequence is Cauchy in $L^2$, so no subsequence converges there. The embedding $H^1({\mathbb{R}}^n)\hookrightarrow L^2({\mathbb{R}}^n)$ is therefore not a compact operator. This is the noncompactness of a Sobolev embedding by translation caused by loss of compactness at infinity.