On the compact embedding of Sobolev space

Question

In dimension three, we know that the Sobolev space $W^{\frac{13}{11},11}(D)$ is compactly embedded into $W^{1,11}(D)$, where $D$ is a bounded domain in $R^3$ with smooth boundary. My question is: Does the dual space $(W^{1,11}(\mathcal {O}))^*$ embed compactly into the dual space $ (W^{\frac{13}{11},11}(\mathcal {O}))^*$?

Answer 1

If $i:X\to Y$ is compact, so is $i^*:Y^*\to X^*$; moreover since here $i$ is dense, $i^*$ is injective.