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

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.