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  "body": "\n## Summary\n\nLet $0 \u003c p \\le \\infty$.\nThe Hardy space $H^p$ of the unit disc is the space of [[holomorphic functions|0NAHXR1FRH]] whose $L^p$ means over the circles $\\abs{z} = r$ stay bounded as $r \\to 1$.\nFor $p \\ge 1$ it is a [[Banach space|06GZ956657]], and $H^2$ is the Hilbert space of power series $\\sum a_n z^n$ with $\\sum \\abs{a_n}^2 \u003c \\infty$.\nBy [[Fatou's theorem|Q2N917JQ6A]], functions in $H^p$ have boundary values almost everywhere on the circle.\nSo $H^p$ can be seen as a subspace of $L^p$ of the circle.\nHardy spaces connect complex analysis with Fourier analysis and operator theory, through [[inner functions|NN8XH5RJYB]], [[Blaschke products|9N6R5ADJXT]] and [[Toeplitz operators|GQ0X3R3WM2]].\n\n## Explanations\n\n### L1 human\n\nTake a holomorphic function on the unit disc and look at its size along circles of radius $r$ around $0$.\nThe function is in a Hardy space if these average sizes stay bounded as the circles approach the rim.\nSo the function may blow up at some rim points, but not too strongly on average.\nFor example, $\\log(1 - z)$ is unbounded near $z = 1$, yet it lies in $H^2$ because its Taylor coefficients $-1/n$ are square-summable.\n",
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