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  "body": "\n## Definition\n\nLet $\\Omega$ be a [[domain|0EWMDD6C14]] in $\\C$, let $dA$ be area measure on $\\Omega$, and let $0 \u003c p \u003c \\infty$.\nThe *Bergman space* $A^p(\\Omega)$ is the set of [[holomorphic functions|0NAHXR1FRH]] $f \\colon \\Omega \\to \\C$ such that\n$$\\norm{f}_{A^p} = \\left( \\int_\\Omega \\abs{f(z)}^p \\, dA(z) \\right)^{1/p} \u003c \\infty.$$\n\n## Remarks\n\nThe sub-mean value property of $\\abs{f}^p$ gives $\\abs{f(z)}^p \\le \\frac{1}{\\pi \\delta^2}\\int_{\\abs{w - z} \u003c \\delta} \\abs{f(w)}^p \\, dA(w)$ whenever the disc of radius $\\delta$ about $z$ lies in $\\Omega$.\nSo convergence in $A^p$ implies locally uniform convergence, and for $p \\ge 1$ the space $A^p(\\Omega)$ is a closed subspace of $L^p(\\Omega)$, hence a Banach space.\nOn the [[unit disc|8G5ABXRK85]] $\\mathbb{D}$, the monomials satisfy $\\int_{\\mathbb{D}} \\abs{z}^{2n}\\, dA = \\int_0^{2\\pi}\\int_0^1 r^{2n+1}\\, dr\\, d\\theta = \\pi/(n + 1)$, so $\\norm{\\sum a_n z^n}_{A^2}^2 = \\pi \\sum_n \\abs{a_n}^2/(n + 1)$.\nExample: $g(z) = 1/(1 - z) = \\sum z^n$ on $\\mathbb{D}$ has $\\abs{g(z)}^p = \\abs{1 - z}^{-p}$, which is area-integrable near $z = 1$ exactly when $p \u003c 2$; so $g \\in A^p(\\mathbb{D})$ for $p \u003c 2$.\nNon-example: $g \\notin A^2(\\mathbb{D})$, as also seen from $\\pi \\sum 1/(n + 1) = \\infty$.\nSome authors use the normalized area measure $dA/\\pi$ on $\\mathbb{D}$, which multiplies the norm by a constant.\n",
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