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  "body": "\n## Statement\n\nLet $X$ be a real [[normed vector space|03J5RAKB30]] and let $K \\subseteq X$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].\nLet $f \\colon K \\to K$ be a [[continuous map|0X3WGX95FE]].\nThen $f$ has a fixed point: there is $x \\in K$ with $f(x) = x$.\n\n## Remarks\n\nThe proof reduces to the Brouwer fixed-point theorem.\nFor $\\varepsilon \u003e 0$, compactness gives a finite $\\varepsilon$-net $\\set{y_1, \\ldots, y_m}$ of $K$.\nThe Schauder projection maps $K$ continuously into the convex hull of the $y_i$ and moves each point by less than $\\varepsilon$.\nBrouwer's theorem gives a fixed point of the projected map on this finite-dimensional convex set, and it is an $\\varepsilon$-fixed point of $f$.\nA limit as $\\varepsilon \\to 0$ gives a fixed point of $f$.\n\nAn equivalent form is often used in analysis.\nLet $C$ be a nonempty closed convex subset of a Banach space and let $f \\colon C \\to C$ be continuous with $f(C)$ contained in a compact set; then $f$ has a fixed point.\n\nExample: Peano's existence theorem for $y' = F(t, y)$, $y(0) = y_0$ with continuous $F$ follows.\nThe integral operator $y \\mapsto y_0 + \\int_0^t F(s, y(s)) \\, ds$ maps a closed convex set of continuous functions into a compact set, by the Arzelà–Ascoli theorem.\nIts fixed point is a solution.\n\nCompactness cannot be dropped: on the closed unit ball of $\\ell^2$, Kakutani's map $x \\mapsto \\big(\\sqrt{1 - \\norm{x}^2}, x_1, x_2, \\ldots\\big)$ is continuous with no fixed point.\n\n## Explanations\n\n### L1 human\n\nBrouwer's theorem says that a continuous map of a disc or ball into itself fixes some point.\nSchauder's theorem extends this to infinite-dimensional spaces, such as spaces of functions, provided the set is compact and convex.\nIt turns many existence problems for differential and integral equations into the search for a fixed point.\n\n### L2 human\n\nThe theorem is the main tool for nonlinear problems without uniqueness, where the Banach fixed point theorem does not apply.\nCompactness usually comes from an Arzelà–Ascoli argument or a compact Sobolev embedding.\nTychonoff extended it to compact convex subsets of locally convex spaces, and [[Schaefer's theorem|7H3YSSDEBK]] replaces the invariant set by an a priori bound.\nFor set-valued maps the analogue is the [[Kakutani fixed-point theorem|7V3HWGR32Z]] and its infinite-dimensional versions.\n",
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