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  "body": "\n## Definition\n\nLet $X$ be a real [[Banach space|06GZ956657]] with [[continuous dual space|4A82S5FVRT]] $X^*$.\nLet $I \\colon X \\to \\R$ be continuously [[Fréchet differentiable|06EH97V2A7]], with derivative $I'(u) \\in X^*$ at $u$.\nA *Palais–Smale sequence* for $I$ is a sequence $(u_n)$ in $X$ such that $(I(u_n))$ is bounded and $\\norm{I'(u_n)}_{X^*} \\to 0$.\nThe functional $I$ satisfies the *Palais–Smale condition* if every Palais–Smale sequence for $I$ has a convergent subsequence.\n\n## Remarks\n\nExample: on a real Hilbert space $H$, let $I(u) = \\tfrac12 \\norm{u}^2$.\nUnder 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.\nNon-example: $I(x) = e^x$ on $\\R$.\nThe sequence $x_n = -n$ has $0 \u003c I(x_n) \\le 1$ and $I'(x_n) = e^{-n} \\to 0$, but no subsequence converges.\nA common variant at a level $c \\in \\R$, written $(PS)_c$, asks only for sequences with $I(u_n) \\to c$ and $I'(u_n) \\to 0$.\nIf $I$ satisfies the condition, then the set of critical points at any level $c$ is compact.\n",
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