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  "body": "\n## Definition\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ be the open [[unit disc|8G5ABXRK85]] and $\\overline{\\mathbb{D}}$ its closure.\nThe *disk algebra* $A(\\mathbb{D})$ is the set of functions $f \\colon \\overline{\\mathbb{D}} \\to \\C$ such that\n\n1. $f$ is [[continuous|0P27W9RW0G]] on $\\overline{\\mathbb{D}}$, and\n2. the restriction of $f$ to $\\mathbb{D}$ is [[holomorphic|0DG0Z72CQD]],\n\nwith pointwise addition and multiplication and the norm $\\norm{f} = \\sup_{z \\in \\overline{\\mathbb{D}}} \\abs{f(z)}$.\n\n## Remarks\n\n$A(\\mathbb{D})$ is a closed subalgebra of $C(\\overline{\\mathbb{D}})$, hence a commutative Banach algebra with unit.\nBy the [[maximum modulus principle|048F2P90WR]], the norm equals $\\sup_{\\abs{z} = 1} \\abs{f(z)}$, so $A(\\mathbb{D})$ may also be viewed as an algebra of functions on the circle.\nThe polynomials are dense: $f(rz) \\to f(z)$ uniformly as $r \\to 1$, and each $f(rz)$ is the uniform limit of its Taylor polynomials on $\\overline{\\mathbb{D}}$.\nThe maximal ideals of $A(\\mathbb{D})$ are the sets $\\set{f : f(a) = 0}$ for $a \\in \\overline{\\mathbb{D}}$.\n\nExample: $f(z) = \\sum_{n \\ge 1} z^n / n^2$.\nThe terms are bounded by $1/n^2$ on $\\overline{\\mathbb{D}}$, so the series converges uniformly there; hence $f$ is continuous on $\\overline{\\mathbb{D}}$ and holomorphic in $\\mathbb{D}$.\n\nNon-example: $S(z) = \\exp\\left(-\\frac{1 + z}{1 - z}\\right)$ is bounded and holomorphic in $\\mathbb{D}$, but it has no continuous extension to $\\overline{\\mathbb{D}}$.\nIts radial limit at $1$ is $0$, while its radial limits at the points $e^{it} \\ne 1$ have modulus $1$, so no continuous extension to $\\overline{\\mathbb{D}}$ exists.\n",
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