The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations.
In finite-dimensional spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically for proper maps: functions which do not take unbounded sets into bounded sets.
In the calculus of variations, where one is typically interested in infinite-dimensional function spaces, the condition is necessary because some extra notion of compactness beyond simple boundedness is needed.
See, for example, the proof of the mountain pass theorem in section 8.5 of Evans.
A continuously Fréchet differentiable functional
from a Hilbert space H to the reals satisfies the Palais–Smale condition if every sequence
such that: has a convergent subsequence in H. Let X be a Banach space and
is said to satisfy the weak Palais–Smale condition if for each sequence