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  "body": "\n## Summary\n\nThe Bloch space is the space of [[holomorphic functions|0NAHXR1FRH]] $f$ on the unit disc for which $(1 - \\abs{z}^2)\\abs{f'(z)}$ stays bounded.\nWith a natural norm it is a [[Banach space|06GZ956657]], and the seminorm $\\sup_z (1 - \\abs{z}^2)\\abs{f'(z)}$ does not change under composition with disc automorphisms.\nIt contains all bounded holomorphic functions and also some unbounded ones such as $\\log(1 - z)$.\nIt is closely tied to [[Bloch's theorem|KW31KDX7TB]], to the space of [[bounded mean oscillation|QJYTXFT8Y4]], and to the duality theory of [[Bergman spaces|2D619WTPYC]].\n\n## Explanations\n\n### L1 human\n\nA function in the Bloch space may grow near the edge of the disc, but its derivative may grow at most like $1/(1 - \\abs{z})$.\nThe factor $1 - \\abs{z}^2$ is the natural scale of the disc in hyperbolic geometry, so the condition says that $f$ changes by a bounded amount over each hyperbolic step.\nFor example, $\\log(1 - z)$ is unbounded near $z = 1$, yet it belongs to the Bloch space.\n",
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