= Solution
Using the usual <zero-boundary Sobolev space> convention,
$$
H_0^1(D)=\overline{C_c^\infty(D)}^{\,H^1(D)},\qquad \|u\|_{H^1(D)}^2=\int_D(|u|^2+|\nabla u|^2)\,dA,
$$
where <weak derivatives> define $H^1(D)$ and $C_c^\infty(D)$ is the <space of test functions>. In the <Gaussian free field> convention, the same notation often denotes the <Dirichlet energy space>, the completion in the <gradient> <norm> alone. On bounded <domains> the <Poincare inequality> makes the two definitions equivalent; on unbounded <domains> one must distinguish them. The following gradient-pairing argument applies in either setting whenever the energy completion is realized as weak functions.
Identify $H_{\mathrm{supp}}=H_0^1(U)$ with a <vector subspace> of $H_0^1(D)$ by <zero extension of H01>. Approximating by <test functions> in $U$ shows that this is an <isometric embedding> in the inhomogeneous <Sobolev norm> and also in the <Dirichlet inner product> <norm>. In particular, arbitrary irregularity of $\partial U$ causes no additional <boundary> term.
Define
$$
H_{\mathrm{harm}}=\{h\in H_0^1(D):\Delta h=0\text{ in distributions on }U\}.
$$
These are <weakly harmonic Sobolev functions>. For $\phi\in C_c^\infty(U)$, <integration by parts> in the weak sense gives $\int_U\nabla h\cdot\nabla\phi\,dA=0$. If $u\in H_{\mathrm{supp}}$, choose $\phi_n\in C_c^\infty(U)$ converging to $u$ in $H^1(U)$, or in energy for the homogeneous convention. The <Cauchy-Schwarz inequality> gives
$$
(h,u)_{\nabla,D}=\lim_{n\to\infty}(h,\phi_n)_{\nabla,D}=0.
$$
Hence the <orthogonality of supported and harmonic Dirichlet functions> is
$$
\boxed{H_{\mathrm{supp}}\perp H_{\mathrm{harm}}.}
$$
Both are linear <vector subspaces>. In the inhomogeneous convention they are closed: the first is the isometric image of a complete space, and the second is the intersection of the kernels of the <linear functionals> $h\mapsto(h,\phi)_\nabla$. No spanning assertion is needed. The PDF contains this <orthogonality> statement; the TeX has badly corrupted it into an assertion about openness.
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