The Sobolev space $H^1$ is a Hilbert space

The pairing is a well-defined inner product: each summand $(D^{\alpha }u,D^{\alpha }v{)}_{L^2}$ is the $L^2$ pairing of the well-defined classes $D^{\alpha }u$ and $D^{\alpha }v$, hence representative-independent; each is linear in the first argument and conjugate-linear in the second over $\mathbb{K}$, and conjugate-symmetric. A finite sum of maps with these properties again has them, so $(u,v{)}_{H^1}$ is well defined on classes, linear in $u$, conjugate-linear in $v$ and conjugate-symmetric. It is positive definite, since $(u,u{)}_{H^1}={\sum }_{\alpha \in {\mathcal{A}}_1}\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{L^2}^2\ge 0$ equals $0$ only when every $D^{\alpha }u=0$, in particular $u=D^{0}u=0$, while $u=0$ plainly gives $0$.

$H_0^1(\mathrm{\Omega })$ is a linear subspace. It contains the zero class, as the zero test function shows. Let $u,v\in H_0^1(\mathrm{\Omega })$ and $a,b\in \mathbb{K}$, and let $\delta >0$. Using the test-function approximation of [F5], choose test functions $\phi ,\psi$ with $\mathrm{\parallel }u-\phi {\mathrm{\parallel }}_{W^{1,2}}<\delta \mathrm{/}(2(\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }+1))$ and $\mathrm{\parallel }v-\psi {\mathrm{\parallel }}_{W^{1,2}}<\delta \mathrm{/}(2(\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }+1))$; then $a\phi +b\psi$ is again a test function, and $\mathrm{\parallel }(au+bv)-(a\phi +b\psi ){\mathrm{\parallel }}_{W^{1,2}}\le \mathrm{\mid }a\mathrm{\mid }\mathrm{\parallel }u-\phi {\mathrm{\parallel }}_{W^{1,2}}+\mathrm{\mid }b\mathrm{\mid }\mathrm{\parallel }v-\psi {\mathrm{\parallel }}_{W^{1,2}}<\delta$. So $au+bv\in H_0^1(\mathrm{\Omega })$; the space is a subspace and, being a closure, it is closed in $H^1(\mathrm{\Omega })$.

Its induced norm is the Sobolev norm: $(u,u{)}_{H^1}={\sum }_{\alpha \in {\mathcal{A}}_1}\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{L^2}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,2}}^2$, so the induced length is $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,2}}$.

Completeness: let $(u_m)$ be a Cauchy sequence in $H^1$. For each $\alpha \in {\mathcal{A}}_1$ the inequality $\mathrm{\parallel }{D}^{\alpha }{u}_m-D^{\alpha }{u}_l{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }{u}_m-u_l{\mathrm{\parallel }}_{W^{1,2}}$ shows that $(D^{\alpha }{u}_m{)}_m$ is Cauchy in $L^2$, so it has a limit class $f_{\alpha }$. Fix $i$ and a test function $\phi$. The weak-derivative identity gives ${\int }_{\mathrm{\Omega }}{u}_m{D}_i\phi dx=-{\int }_{\mathrm{\Omega }}{D}_i{u}_m\phi dx$ for every $m$; H"older's inequality makes both sides converge to ${\int }_{\mathrm{\Omega }}{f}_0{D}_i\phi dx$ and $-{\int }_{\mathrm{\Omega }}{f}_i\phi dx$, respectively, where $f_0$ is the limit of the classes $u_m=D^0{u}_m$. Hence $\int f_0{D}_i\phi =-\int f_i\phi$ for every test function $\phi$, so $f_0\in W^{1,2}(\mathrm{\Omega };\mathbb{K})$ with $D_i{f}_0=f_i$, and $\mathrm{\parallel }{u}_m-f_0{\mathrm{\parallel }}_{W^{1,2}}^2={\sum }_{\alpha \in {\mathcal{A}}_1}\mathrm{\parallel }{D}^{\alpha }{u}_m-f_{\alpha }{\mathrm{\parallel }}_{L^2}^2\to 0$. Thus every Cauchy sequence in $H^1$ converges in $H^1$: the space is complete in its Sobolev norm.

Consequences for the pairing: by steps 1.1, 2.1 the pairing is an inner product inducing the Sobolev norm, and by step 3.1 the space is complete for that norm; therefore $H^1(\mathrm{\Omega })$ is a Hilbert space over $\mathbb{K}$ for the pairing.

$H_0^1(\mathrm{\Omega })$ is a Hilbert space: it is a closed linear subspace of the Hilbert space $H^1(\mathrm{\Omega })$, hence complete for the restricted norm by [F6] applied to the underlying Banach space, and the restriction of the inner product is an inner product whose induced norm is the restriction of the Sobolev norm.

The pairing $(\cdot ,\cdot {)}_{H^1}$ is therefore an inner product on $H^1(\mathrm{\Omega })$ inducing the $W^{1,2}$ norm, making $H^1(\mathrm{\Omega })$ a Hilbert space, while $H_0^1(\mathrm{\Omega })$ is a closed subspace and a Hilbert space for the restricted pairing; the pairing is linear in the first argument and conjugate-linear in the second, as required.