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  "body": "\n## Definition\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ be the [[unit disc|8G5ABXRK85]] and let $dA$ be area measure.\nThe *Dirichlet space* $\\mathcal{D}$ is the set of [[holomorphic functions|0NAHXR1FRH]] $f \\colon \\mathbb{D} \\to \\C$ such that\n$$\\int_{\\mathbb{D}} \\abs{f'(z)}^2 \\, dA(z) \u003c \\infty.$$\nIts norm is given by $\\norm{f}_{\\mathcal{D}}^2 = \\abs{f(0)}^2 + \\frac{1}{\\pi}\\int_{\\mathbb{D}} \\abs{f'(z)}^2 \\, dA(z)$.\n\n## Remarks\n\nFor $f(z) = \\sum_{n \\ge 0} a_n z^n$, a [[power series|0D6KXBHKC0]] on $\\mathbb{D}$, one has $\\int_{\\mathbb{D}} \\abs{z^{n-1}}^2 dA = \\pi/n$, hence $\\int_{\\mathbb{D}} \\abs{f'}^2 \\, dA = \\pi\\sum_{n \\ge 1} n \\abs{a_n}^2$ and $\\norm{f}_{\\mathcal{D}}^2 = \\abs{a_0}^2 + \\sum_{n \\ge 1} n\\abs{a_n}^2$.\nThis norm comes from an [[inner product|0QVNCJXRY3]], and $\\mathcal{D}$ is a Hilbert space.\nExample: $f(z) = z$ has $\\int_{\\mathbb{D}} \\abs{f'}^2 \\, dA = \\pi$, the area of $f(\\mathbb{D}) = \\mathbb{D}$.\nNon-example: $f(z) = \\log(1 - z) = -\\sum_{n \\ge 1} z^n/n$ gives $\\sum_{n \\ge 1} n \\cdot \\frac{1}{n^2} = \\sum 1/n = \\infty$, so $f \\notin \\mathcal{D}$, although $f$ is in the Hardy space $H^2$.\nIf $f$ is injective, $\\int_{\\mathbb{D}} \\abs{f'}^2 \\, dA$ is the area of $f(\\mathbb{D})$, since $\\abs{f'}^2$ is the Jacobian determinant of $f$.\nSome authors omit the term $\\abs{f(0)}^2$ or replace it by $\\norm{f}_{H^2}^2$; these give equivalent norms.\n",
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