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  "body": "\n## Definition\n\nLet $\\Omega$ be a [[domain|0EWMDD6C14]] in $\\C$, and let $A^2(\\Omega)$ be its [[Bergman space|GZV7QFA0RZ]] with the [[inner product|0QVNCJXRY3]] $\\ip{f}{g} = \\int_\\Omega f \\bar g \\, dA$.\nFor each $w \\in \\Omega$, the map $f \\mapsto f(w)$ is a bounded linear functional on the Hilbert space $A^2(\\Omega)$.\nBy the Riesz representation theorem there is a unique $K_w \\in A^2(\\Omega)$ with\n$$f(w) = \\ip{f}{K_w} = \\int_\\Omega f(z)\\, \\overline{K_w(z)}\\, dA(z) \\quad \\text{for all } f \\in A^2(\\Omega).$$\nThe *Bergman kernel* of $\\Omega$ is the function $K \\colon \\Omega \\times \\Omega \\to \\C$ given by $K(z, w) = K_w(z)$.\n\n## Remarks\n\nThe kernel satisfies $K(w, z) = \\overline{K(z, w)}$, and $K(w, w) = \\norm{K_w}^2 \u003e 0$ when $A^2(\\Omega) \\ne \\set{0}$.\nFor any [[orthonormal basis|0S9GGTR0EX]] $(e_n)$ of $A^2(\\Omega)$, $K(z, w) = \\sum_n e_n(z)\\overline{e_n(w)}$.\nExample: on the unit disc the functions $e_n(z) = \\sqrt{(n + 1)/\\pi}\\, z^n$ form an orthonormal basis, so $K(z, w) = \\frac{1}{\\pi}\\sum_{n \\ge 0} (n + 1)(z\\bar w)^n = \\frac{1}{\\pi(1 - z\\bar w)^2}$.\nNon-example: the function $\\frac{1}{1 - z\\bar w}$ is not the Bergman kernel of the disc.\nIt is the Szegő kernel, which reproduces values for the inner product of the Hardy space $H^2$ with normalized arc length, not for the area inner product.\nFor instance, with $w = 0$ it is the constant $1$, while $\\int_{\\mathbb{D}} f \\cdot 1 \\, dA = \\pi f(0)$, not $f(0)$.\nIf $\\varphi \\colon \\Omega_1 \\to \\Omega_2$ is biholomorphic, then $K_1(z, w) = \\varphi'(z) K_2(\\varphi(z), \\varphi(w)) \\overline{\\varphi'(w)}$.\n",
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