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  "body": "\n## Summary\n\nThe Palais–Smale condition is a compactness hypothesis on a continuously differentiable functional on a Banach space.\nIt asks that every sequence along which the functional stays bounded and its derivative tends to zero has a convergent subsequence.\nIt replaces the compactness of closed bounded sets, which fails in infinite dimensions, in the search for [[critical points|0YTCMFB4HC]].\nIt is the main hypothesis of the mountain pass theorem and of many other minimax results.\n\n## Explanations\n\n### L1 human\n\nTo find a lowest point or a mountain pass of a landscape, one usually walks along points where the ground gets flatter and flatter.\nIn infinite dimensions such a walk can run away to infinity or spread out without ever reaching a point.\nThe Palais–Smale condition forbids this: every such walk with bounded heights must have a part that settles at an actual point.\nFor example, $f(x) = e^x$ on $\\R$ fails it, since its slope tends to $0$ as $x \\to -\\infty$ while no flat point exists.\n",
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