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  "body": "\n## Definition\n\nLet $M$ be a $C^\\infty$ [[Hilbert manifold|R8DMMW0E9M]] with a complete Riemannian metric $g$, and let $f \\colon M \\to \\R$ be a $C^2$ function.\nThe gradient $\\nabla f$ is defined by $g(\\nabla f, v) = df(v)$.\nThe function $f$ satisfies *condition (C)* if every sequence $(p_n)$ in $M$ on which $f$ is bounded and $\\norm{\\nabla f(p_n)} \\to 0$ has a convergent subsequence.\nThis is the [[Palais–Smale condition|4724KRNHAT]] on $M$.\nThrough $g$, the Hessian of $f$ at a critical point $p$ is a bounded self-adjoint operator on $T_p M$.\nThe point $p$ is *nondegenerate* if this operator is invertible.\nIts *index* is the supremum of the dimensions of subspaces of $T_p M$ on which the Hessian is negative definite.\n\n*Morse theory on Hilbert manifolds* is the [[critical point theory|9ZQC7ARWTT]] of $C^2$ functions $f$ on such $M$ with three properties.\nThe function is bounded below, it satisfies condition (C), and its critical points are [[nondegenerate|MGPPCDFQDG]] of finite [[index|MAX2ZYG6H9]].\n\n## Remarks\n\nFor such $f$ the two basic theorems of Morse theory hold.\nIf $a \u003c b$ and $f^{-1}[a, b]$ contains no critical point, then the sublevel set $M^a = \\set{f \\le a}$ is a deformation retract of $M^b$.\nIf $f^{-1}[a, b]$ contains exactly one critical point, of index $\\lambda$, in its interior, then $M^b$ has the homotopy type of $M^a$ with a $\\lambda$-cell attached.\n\nExample: let $(N, h)$ be a compact Riemannian manifold, let $p, q \\in N$, and let $M$ be the Hilbert manifold of paths $\\omega \\colon [0, 1] \\to N$ of Sobolev class $H^1$ with $\\omega(0) = p$ and $\\omega(1) = q$.\nThe [[energy|Q12AB1MJ64]] $E(\\omega) = \\frac{1}{2} \\int_0^1 \\abs{\\omega'}^2 \\, dt$ is bounded below and satisfies condition (C).\nIts critical points are the geodesics from $p$ to $q$, of finite index, and they are nondegenerate when $q$ is not conjugate to $p$ along any geodesic.\n\nNon-example: on $M = \\R$ the function $f(x) = \\arctan x$ is bounded below and has no critical points, but condition (C) fails along $x_n = -n$, since $f(x_n)$ is bounded and $f'(x_n) \\to 0$.\nAccordingly $f$ does not attain its infimum $-\\pi/2$, while a function that is bounded below and satisfies condition (C) attains its infimum at a critical point.\n",
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