I'm reading a hand-waving argument in a proof of Chapter 7 of the *Navier-Stokes Equations* by Constantin and Foias. I would like to know if I understand it correctly.

Let $\Omega\subset{\mathbb{R}^n}$ be an open set with $\partial \Omega$ being $C^k$, $k\geq 2$. Let $\mathcal{V}$ be the space

$$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot u=0\}.$$

Let $ H=\overline{\mathcal{V}}^{\|\cdot\|_{L^2(\Omega)^n}} $ and $ V=\overline{\mathcal{V}}^{\|\cdot\|_{H_0^1(\Omega)^n}}. $

Is the following statement true?

Suppose $u_m\to u$ weakly in $V$. Then there exist a subsequence $u_{m'}\to u$ strongly in $H$.

compact. This weak-to-strong trick is just a frequently helpful little extra fact. The key point is the compactness, as @MichaelRenardy notes. $\endgroup$ – paul garrett Jun 13 '16 at 22:27