Palais–Smale compactness condition

Summary

The Palais–Smale condition is a compactness hypothesis on a continuously differentiable functional on a Banach space. It asks that every sequence along which the functional stays bounded and its derivative tends to zero has a convergent subsequence. It replaces the compactness of closed bounded sets, which fails in infinite dimensions, in the search for critical points. It is the main hypothesis of the mountain pass theorem and of many other minimax results.