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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be open, let $k \\ge 0$ be an integer, and let $0 \u003c \\alpha \\le 1$.\nFor a function $v \\colon U \\to \\R$, its Hölder seminorm of exponent $\\alpha$ is\n$$[v]_{\\alpha, U} = \\sup_{x, y \\in U, \\; x \\ne y} \\frac{\\abs{v(x) - v(y)}}{\\abs{x - y}^{\\alpha}}.$$\nThe *Hölder space* $C^{k,\\alpha}(\\overline{U})$ is the set of functions $u \\colon U \\to \\R$ such that:\n\n1. $u$ is $k$ times continuously differentiable on $U$;\n2. every [[partial derivative|09EGC6F8PT]] $\\partial^\\beta u$ with $\\abs{\\beta} \\le k$ is bounded on $U$;\n3. every partial derivative $\\partial^\\beta u$ with $\\abs{\\beta} = k$ satisfies $[\\partial^\\beta u]_{\\alpha, U} \u003c \\infty$.\n\nIts norm is\n$$\\norm{u}_{C^{k,\\alpha}(\\overline{U})} = \\sum_{\\abs{\\beta} \\le k} \\sup_{U} \\abs{\\partial^\\beta u} + \\sum_{\\abs{\\beta} = k} [\\partial^\\beta u]_{\\alpha, U}.$$\n\n## Remarks\n\nWith this norm, $C^{k,\\alpha}(\\overline{U})$ is a [[Banach space|06GZ956657]].\nCondition 3 says that the derivatives of order $k$ are [[Hölder continuous|012PBA0FMD]] with exponent $\\alpha$ on $U$, so they extend continuously to the closure $\\overline{U}$.\nThe space $C^{0,1}(\\overline{U})$ is the space of bounded Lipschitz functions.\nOn a bounded set $U$, $C^{k,\\beta}(\\overline{U}) \\subseteq C^{k,\\alpha}(\\overline{U})$ for $0 \u003c \\alpha \u003c \\beta \\le 1$.\nThe local space $C^{k,\\alpha}(U)$ is the set of $u$ with $u \\in C^{k,\\alpha}(\\overline{V})$ for every open $V$ with compact closure in $U$.\n\nExample: $u(x) = \\sqrt{\\abs{x}}$ lies in $C^{0,1/2}([-1, 1])$.\nIt is continuous and bounded by $1$, and $\\abs{\\sqrt{\\abs{x}} - \\sqrt{\\abs{y}}} \\le \\sqrt{\\abs{\\abs{x} - \\abs{y}}} \\le \\abs{x - y}^{1/2}$.\n\nNon-example: the same $u$ is not in $C^{0,1}([-1, 1])$, so it fails condition 3 for $\\alpha = 1$.\nIndeed $\\abs{u(x) - u(0)} / \\abs{x - 0} = \\abs{x}^{-1/2}$ is unbounded as $x \\to 0$.\n",
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