A Universe of Sorts

The theorem states that given a system $B(u, -) = f(-)$ , where $B$ is a linear, bounded, coercive operator, then a unique solution exists and this solution depends continuously on $f$ .
This is useful to solve elliptic equation problems, for which one can find some kind of inner product $B(., .)$ that represents "energy".

§ Lax Milgram Theorem