Origin and evolution of the Palais–Smale condition in critical point theory

In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C1 defined on a Riemannian manifold modeled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais–Smale-type conditions. The paper describes their evolution.