= Sobolev inner product
{c}
{title2=$(u,v)_{H^1}=\int_\Omega uv+\nabla u\cdot\nabla v$}
For real-valued <functions>, the <inner product>
$$
(u,v)_{H^1}=\int_\Omega uv+\nabla u\cdot\nabla v
$$
on the <H1 space> induces its usual <Sobolev norm>. For complex-valued <functions>, use $u\overline v+\nabla u\cdot\overline{\nabla v}$ under the convention that the <inner product> is linear in its first argument. Omitting conjugation in the complex setting would fail positivity, for example on a purely imaginary constant function.
A real reaction-diffusion <weak formulation> with unit diffusion and unit reaction has this exact <bilinear form>. Its conforming <Galerkin method> is therefore an <orthogonal projection> and gives a <best approximation> with constant one in the first-order <Sobolev norm>.
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