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  "body": "\n## Statement\n\nLet $M$ be a $C^\\infty$ [[Hilbert manifold|R8DMMW0E9M]] with a complete Riemannian metric.\nLet $f \\colon M \\to \\R$ be a $C^2$ function that is bounded below and satisfies the [[Palais–Smale condition|4724KRNHAT]].\nThis means that every sequence $(p_n)$ with $(f(p_n))$ bounded and $\\norm{\\nabla f(p_n)} \\to 0$ has a convergent subsequence.\nFor $c \\in \\R$ let $M^c = \\set{p \\in M : f(p) \\le c}$.\n\n1. (Deformation) If $a \u003c b$ and $f^{-1}[a, b]$ contains no critical point of $f$, then $M^a$ is a strong [[deformation retract|YW8HBW1GJX]] of $M^b$.\n2. (Minimum) The infimum of $f$ is attained, and every point where it is attained is a critical point of $f$.\n3. (Lusternik–Schnirelmann bound) The number of critical points of $f$ is at least $\\operatorname{cat}(M) + 1$, where $\\operatorname{cat}$ is the normalized [[Lusternik–Schnirelmann category|W5BHHXCYXA]].\n\n## Remarks\n\nThe proof of the deformation statement follows the flow of $-\\nabla f / \\norm{\\nabla f}^2$ on $f^{-1}[a, b]$.\nThe Palais–Smale condition gives a positive lower bound for $\\norm{\\nabla f}$ there, and completeness of the metric lets the flow run until it reaches $M^a$.\nFor the minimum, suppose $c = \\inf f$ were not a critical value.\nThen the condition would give $\\varepsilon \u003e 0$ with no critical value in $[c, c + \\varepsilon]$.\nThe same flow, extended below $c$, would push the nonempty set $M^{c + \\varepsilon}$ into a sublevel set $M^{c'}$ with $c' \u003c c$, which is empty.\nFor the third statement, the minimax values $c_k = \\inf \\set{\\sup_A f : A \\subseteq M \\text{ closed}, \\ \\operatorname{cat}_M(A) \\ge k}$ for $k = 0, \\dots, \\operatorname{cat}(M)$ are critical values, as in [[Lusternik–Schnirelmann theory|QARH15D2PN]], and equal values force infinitely many critical points.\n\nExample: the energy functional on paths with fixed endpoints in a compact Riemannian manifold, or on its free loop space, satisfies these hypotheses.\nThis gives existence of geodesics and is the setting of the [[Lyusternik–Fet theorem|EXZA8ZA8P6]].\nWithout the Palais–Smale condition the conclusions fail: $f(x) = e^x$ on $\\R$ is bounded below and has no critical point, so its infimum $0$ is not attained.\nThe [[mountain pass theorem|EPXNCSP2MN]] is another critical point theorem that rests on the same condition.\n\n## Explanations\n\n### L1 human\n\nOn a compact manifold, every continuous function has a minimum, and topology forces further critical points.\nIn infinite dimensions this fails, because a minimizing sequence can run off to infinity.\nPalais and Smale found one condition on the function that restores what is needed: almost-critical sequences must have convergent subsequences.\nWith it, the classical statements about minima, deformations and the number of critical points hold again.\n\n### L2 human\n\nThese results are the base of variational methods for nonlinear partial differential equations and for geodesics.\nTo apply them, one checks the Palais–Smale condition for a concrete functional, typically with Sobolev embeddings and compactness theorems.\nThe deformation statement is the engine of all minimax arguments, including the [[mountain pass theorem|EPXNCSP2MN]] and the Lusternik–Schnirelmann bound.\nThe same results hold for $C^1$ functions on complete Finsler manifolds modelled on Banach spaces, with a pseudo-gradient vector field in place of the gradient.\n",
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