Lax-Milgram Theorem

A functional-analytic result asserting that if a(·,·) is a continuous (bounded) bilinear form on a Hilbert space H that is coercive (there exists α>0 with a(v,v) ≥ α||v||^2 for all v), then for every continuous linear functional f there exists a unique u in H satisfying a(u,v) = f(v) for all v. It provides existence, uniqueness, and a priori estimates for variational formulations of PDEs.