Sobolev inner product ($(u,v{)}_{H^1}={\int }_{\Omega }uv+\nabla u\cdot \nabla v$)

For real-valued functions, the inner producton 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.