---
tag: KA3V66S6F5
kind: definition
name: Palais–Smale compactness condition
msc: ["58E05", 47J30]
defines: 4724KRNHAT
uses: [06GZ956657, 06EH97V2A7, 4A82S5FVRT]
sources:
  - url: https://en.wikipedia.org/wiki/Palais–Smale_compactness_condition
---

## Definition

Let $X$ be a real [[Banach space|06GZ956657]] with [[continuous dual space|4A82S5FVRT]] $X^*$.
Let $I \colon X \to \R$ be continuously [[Fréchet differentiable|06EH97V2A7]], 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.