Coercive functional

Summary

A functional on a normed space is coercive if its values tend to $+\infty$ as the norm of the argument tends to infinity. Then every minimising sequence is bounded, which is the compactness step of the direct method. In reflexive spaces bounded sequences have weakly convergent subsequences, so coercivity and weak lower semicontinuity together give a minimiser. The word is related to, but different from, a coercive bilinear form.