Global analysis, analysis on manifolds Definition

Palais–Smale compactness condition

Builds on Banach space Fréchet derivative Continuous dual space

Definition

Let $X$ be a real Banach space with continuous dual space $X^*$ . Let $I \colon X \to \R$ be continuously Fréchet differentiable, with derivative $I'(u) \in X^*$ at $u$ . A Palais–Smale sequence for $I$ is a sequence $(u_n)$ in $X$ such that $(I(u_n))$ is bounded and $\norm{I'(u_n)}_{X^*} \to 0$ . The functional $I$ satisfies the Palais–Smale condition if every Palais–Smale sequence for $I$ has a convergent subsequence.

Remarks

Example: on a real Hilbert space $H$ , let $I(u) = \tfrac12 \norm{u}^2$ . Under the identification of $H^*$ with $H$ , the derivative is $I'(u) = u$ , so a Palais–Smale sequence satisfies $u_n \to 0$ , and the condition holds. Non-example: $I(x) = e^x$ on $\R$ . The sequence $x_n = -n$ has $0 < I(x_n) \le 1$ and $I'(x_n) = e^{-n} \to 0$ , but no subsequence converges. A common variant at a level $c \in \R$ , written $(PS)_c$ , asks only for sequences with $I(u_n) \to c$ and $I'(u_n) \to 0$ . If $I$ satisfies the condition, then the set of critical points at any level $c$ is compact.

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Banach space concept
Continuous dual space concept
Euclidean space object
Total derivative concept
Fréchet derivative concept
Palais–Smale compactness condition definition

Tag	`KA3V66S6F5`
URI	`m2://KA3V66S6F5`
Kind	definition
Writing	W0 stub
MSC	58E05, 47J30
Defines	Palais–Smale compactness condition
Uses	Banach space, Fréchet derivative, Continuous dual space
Changed	2 November 2026

Palais–Smale compactness condition

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