Integer-order Sobolev spaces are Banach

By [F8], AC gives Countable Choice, so the completeness interfaces [F3] and [F4] and the finite-measure property [F7] are available. The index set ${\mathcal{A}}_k$ of [F1] is finite. For every $\alpha \in {\mathcal{A}}_k$ the classes $D^{\alpha }{u}_j$ exist, and comparing the displayed norm formula with the corresponding real or complex $L^p$ norm of one coordinate gives $\mathrm{\parallel }{D}^{\alpha }{u}_j-D^{\alpha }{u}_l{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le \mathrm{\parallel }{u}_j-u_l{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$ for the finite-$p$ sum and for the $p=\mathrm{\infty }$ maximum alike. Hence $(D^{\alpha }{u}_j{)}_j$ is Cauchy in $L^p(\mathrm{\Omega };\mathbb{K})$ for every $\alpha$, and $(u_j{)}_j$ is Cauchy in $L^p(\mathrm{\Omega };\mathbb{K})$ because $D^0{u}_j=u_j$.

By the completeness results [F3] for $\mathbb{K}=\mathbb{R}$ and [F4] for $\mathbb{K}=\mathbb{C}$, each coordinate Cauchy sequence converges in $L^p(\mathrm{\Omega };\mathbb{K})$: there are classes $v_{\alpha }\in L^p(\mathrm{\Omega };\mathbb{K})$ with $D^{\alpha }{u}_j⟶v_{\alpha }\text{in }{L}^p(\mathrm{\Omega };\mathbb{K}),\alpha \in {\mathcal{A}}_k\mathrm{.}$ Put $u:=v_0$, so $u_j\to u$ in $L^p(\mathrm{\Omega };\mathbb{K})$. Each class $v_{\alpha }$ has some measurable representative, and ${\mathcal{A}}_k$ is finite, so choosing one representative per coordinate is a finite selection, provable without any choice principle.

Fix $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and put $K=\mathrm{supp}\phi$. By [F7], $K$ has finite measure, and both $\phi$ and $D^{\alpha }\phi$ are bounded and supported in $K$ for every $\alpha$. For each $j$ and $\alpha$, [F1] gives the weak identity ${\int }_{\mathrm{\Omega }}{u}_j{D}^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}(D^{\alpha }{u}_j)\phi dx\mathrm{.}$ Since $D^{\alpha }{u}_j\to v_{\alpha }$ and $u_j\to u$ in $L^p(\mathrm{\Omega })$, restriction to $K$ gives norm convergence in $L^p(K)$, and Hölder [F5] for real and [F6] for complex representatives, including the endpoint pairs, yields $\mid {\int }_{\mathrm{\Omega }}(u_j-u)D^{\alpha }\phi dx\mid \le \mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_{L^p(K)}\mathrm{\parallel }{D}^{\alpha }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(K)}⟶0$ and $\mid {\int }_{\mathrm{\Omega }}(D^{\alpha }{u}_j-v_{\alpha })\phi dx\mid \le \mathrm{\parallel }{D}^{\alpha }{u}_j-v_{\alpha }{\mathrm{\parallel }}_{L^p(K)}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(K)}⟶0.$ Passing to the limit in the identity gives ${\int }_{\mathrm{\Omega }}u{D}^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}{v}_{\alpha }\phi dx\mathrm{.}$

The test function in step 3.1 was arbitrary and each $v_{\alpha }$ is an $L^p$ class, so the displayed identity exhibits $v_{\alpha }$ as a weak $\alpha$-derivative of $u$ in the sense of [F1]; hence $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$ and $D^{\alpha }u=v_{\alpha }$ for every $\alpha \in {\mathcal{A}}_k$.

Finally, $\mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}\to 0$: for finite $p$ the displayed norm of the difference is the $p$-th root of the finite sum of the numbers $\mathrm{\parallel }{D}^{\alpha }{u}_j-v_{\alpha }{\mathrm{\parallel }}_{L^p}^p$, each of which tends to $0$, and for $p=\mathrm{\infty }$ it is the maximum of the finitely many numbers $\mathrm{\parallel }{D}^{\alpha }{u}_j-v_{\alpha }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$, which likewise tends to $0$. Therefore every Cauchy sequence in $W^{k,p}(\mathrm{\Omega };\mathbb{K})$ converges in its norm, and by [F2] that norm makes $W^{k,p}(\mathrm{\Omega };\mathbb{K})$ a normed space.

The boundary cases are included. For $k=0$ we have ${\mathcal{A}}_0=\{0\}$, so the argument reduces to the single convergence $u_j\to u$ and is exactly the completeness used in step 2.1. If $\mathrm{\Omega }=\mathrm{\varnothing }$, then every $L^p$ class is zero by [F1], the sum over ${\mathcal{A}}_k$ is a finite sum of zeros, and the unique class is its own limit. The endpoints $p=1$ and $p=\mathrm{\infty }$ were handled in steps 1.1 and 3.1 through the endpoint clauses of [F5] and [F6]. The only choice used is Countable Choice, obtained from AC by [F8]; no subsequence or pointwise convergence of the full sequence is used. $\mathrm{\square }$