Proposition 20

The Sobolev spaces $W^{k,p}(U)$ and $W^{k,p}_0$ (U) are Banach spaces of countable type. In particular, the spaces $H^k(U)$ and $H^k_0(U)$ are Hilbert spaces of countable type.

It is enough to prove the assertions concerning $W^{k,p}(U)$.

Let I be the set of $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant k$. The linear mapping $u$ of $W^{k,p}(U)$ into $L^p(U)^I$ which associates with $f$ the family $(\partial^{\alpha}f)_{\alpha\in I}$ is injective; it is continuous and strict by definition of the norm on $W^{k,p}(U)$. To prove that $W^{k,p}(U)$ is complete, it is enough to prove that its image under $u$ is closed. Now let $(f_n)_{n\in\mathbf{N}}$ be a sequence in $W^{k,p}(U)$ such that $(u(f_n))_{n\in\mathbf{N}}$ converges. Let $(g_{\alpha})_{\alpha\in I}\in L^p(U)^I$ be its limit. For $\alpha \in I$, the sequence $(\partial^{\alpha}f_n)_{n\in\mathbf{N}}$ converges in $L^p(U)$ to $g_{\alpha}$. A fortiori, the convergence takes place in $\mathscr{D}'(U)$. Put $f=g_0$. For every $\varphi \in \mathscr{D}$(U), one has

= lim$_{n\rightarrow+\infty}(-1)^{|\alpha|}\langle f_n, \partial^{\alpha}\varphi \rangle =$ lim$_{n\rightarrow+\infty}\langle \partial^{\alpha}f_n, \varphi \rangle =\langle g_{\alpha}, \varphi \rangle$.

This proves that $g_{\alpha}=\partial^{\alpha}f$ for every $\alpha \in I$, hence $(g_{\alpha})_{\alpha\in I}=u(f)$ belongs to the image of $u$.

The space $W^{k,p}(U)$ is identified by $u$ with a subspace of the space $L^p(U)^I$; the latter is of countable type (Proposition 2 of IV, p. 180 and TG, IX, p. 19, Corollary, (ii))), hence the same is true of $W^{k,p}(U) ($loc. cit., (i)).