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  "body": "\n## Definition\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ be the [[unit disc|8G5ABXRK85]].\nThe *Bloch space* $\\mathcal{B}$ is the set of [[holomorphic functions|0NAHXR1FRH]] $f \\colon \\mathbb{D} \\to \\C$ such that\n$$\\sup_{z \\in \\mathbb{D}} (1 - \\abs{z}^2) \\abs{f'(z)} \u003c \\infty.$$\nIts norm is $\\norm{f}_{\\mathcal{B}} = \\abs{f(0)} + \\sup_{z \\in \\mathbb{D}} (1 - \\abs{z}^2) \\abs{f'(z)}$.\n\n## Remarks\n\nWith this norm, $\\mathcal{B}$ is a [[Banach space|06GZ956657]].\nExample: let $f(z) = \\log(1 - z)$ with the principal branch, so $f'(z) = -1/(1 - z)$.\nSince $\\abs{1 - z} \\ge 1 - \\abs{z}$, one has $(1 - \\abs{z}^2)\\abs{f'(z)} \\le (1 - \\abs{z})(1 + \\abs{z})/(1 - \\abs{z}) \u003c 2$, so $f \\in \\mathcal{B}$, although $f$ is unbounded.\nNon-example: $g(z) = 1/(1 - z)$ has $g'(z) = 1/(1 - z)^2$, and for real $z = r \\in (0, 1)$ one gets $(1 - r^2)/(1 - r)^2 = (1 + r)/(1 - r) \\to \\infty$, so $g \\notin \\mathcal{B}$.\nEvery bounded holomorphic function is in $\\mathcal{B}$: if $\\abs{f} \\le 1$, the [[Schwarz–Pick theorem|V1QYA9RW73]] gives $(1 - \\abs{z}^2)\\abs{f'(z)} \\le 1$.\nThe closure of the polynomials in $\\mathcal{B}$ is the little Bloch space $\\mathcal{B}_0$, the set of $f$ with $(1 - \\abs{z}^2)\\abs{f'(z)} \\to 0$ as $\\abs{z} \\to 1$.\n",
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