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  "body": "\n## Definition\n\nLet $X$ be a real [[Banach space|06GZ956657]] and let $\\Omega \\subseteq X$ be a [[bounded|09QSZVVM26]] open set.\nLet $K \\colon \\overline{\\Omega} \\to X$ be a compact map, that is, continuous with $K(\\overline{\\Omega})$ relatively compact.\nLet $y \\in X$ with $y \\notin (I - K)(\\partial\\Omega)$, where $I$ is the identity, and put $r = \\inf_{u \\in \\partial\\Omega} \\norm{u - K(u) - y}$, which is positive.\nChoose a continuous map $K_\\varepsilon \\colon \\overline{\\Omega} \\to X$ with values in a finite-dimensional subspace $E \\subseteq X$ that contains $y$, such that $\\norm{K(u) - K_\\varepsilon(u)} \u003c r$ for all $u \\in \\overline{\\Omega}$.\nLet $\\deg_B$ denote the Brouwer degree of continuous maps on closures of bounded open subsets of the finite-dimensional space $E$.\nThe *Leray–Schauder degree* of $I - K$ on $\\Omega$ at $y$ is\n$$\\deg_{LS}(I - K, \\Omega, y) = \\deg_B\\big((I - K_\\varepsilon)|_{\\overline{\\Omega \\cap E}}, \\, \\Omega \\cap E, \\, y\\big).$$\n\n## Remarks\n\nSuch $K_\\varepsilon$ exist because $K(\\overline{\\Omega})$ is relatively compact, and the right-hand side does not depend on the choice of $K_\\varepsilon$ and $E$; this is what makes the definition sound.\nThe degree has these properties.\n\n1. Normalisation: $\\deg_{LS}(I, \\Omega, y) = 1$ if $y \\in \\Omega$.\n2. Existence: if $\\deg_{LS}(I - K, \\Omega, y) \\ne 0$, then $u - K(u) = y$ for some $u \\in \\Omega$.\n3. Homotopy invariance: if $H \\colon \\overline{\\Omega} \\times [0, 1] \\to X$ is compact and $u - H(u, t) \\ne y$ for all $u \\in \\partial\\Omega$ and $t \\in [0, 1]$, then $\\deg_{LS}(I - H(\\cdot, t), \\Omega, y)$ does not depend on $t$.\n\nExample: let $\\Omega$ be the open unit ball and let $K$ be compact with $\\norm{K(u)} \u003c 1$ on $\\overline{\\Omega}$.\nThe homotopy $H(u, t) = tK(u)$ has no solution of $u = tK(u)$ on the unit sphere, so $\\deg_{LS}(I - K, \\Omega, 0) = \\deg_{LS}(I, \\Omega, 0) = 1$, and $K$ has a fixed point in $\\Omega$.\n\nNon-example: if $X$ is infinite-dimensional, the map $K = -I$ on the closed unit ball is not compact, since that ball is not compact.\nSo this construction does not define a degree for $I - K = 2I$.\n\n## Explanations\n\n### L1 human\n\nThe Leray–Schauder degree counts solutions of $u - K(u) = y$ with signs, for compact $K$, by approximating $K$ by maps into finite-dimensional spaces and using the finite-dimensional degree there.\n",
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