= Orthogonality of supported and harmonic Dirichlet functions
{title2=$H_{\mathrm{supp}}\perp H_{\mathrm{harm}}$}
Let $U\subseteq D$ be open. Regard $H_{\mathrm{supp}}=H_0^1(U)$ as functions on $D$ by <zero extension of H01>, and let $H_{\mathrm{harm}}$ consist of the <weakly harmonic Sobolev functions> on $U$ belonging to $H_0^1(D)$. For $h\in H_{\mathrm{harm}}$, the <Dirichlet inner product> $(h,\phi)_\nabla$ vanishes for each <test function> supported in $U$. Approximate any $u\in H_{\mathrm{supp}}$ by those <test functions> in the <gradient> <norm> and use the <Cauchy-Schwarz inequality>. This gives $\boxed{(h,u)_\nabla=0}$. The proof also works with the inhomogeneous <zero-boundary Sobolev space> convention, and with a homogeneous completion realized as weak functions.
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