Sobolev inner product
= Sobolev inner product
{c}
{title2=$(u,v)_{H^1}=\int_\Omega uv+\nabla u\cdot\nabla v$}
The displayed <inner product> on the first-order <Sobolev space> induces its usual <Sobolev norm>. A continuous reaction-diffusion <weak formulation> with unit diffusion and unit reaction has this exact <bilinear form>, so its conforming <Galerkin method> gives a <best approximation> with constant one in the first-order <Sobolev norm>.