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  "body": "\n## Description\n\nLet $n \\ge 1$.\nThe *unit ball* of $\\C^n$ is\n$$\\mathbb{B}^n = \\set{z \\in \\C^n : \\abs{z_1}^2 + \\cdots + \\abs{z_n}^2 \u003c 1}.$$\nIt is the [[open ball|0YE5868HZB]] of radius $1$ about $0$ for the norm $\\abs{z} = (\\sum_j \\abs{z_j}^2)^{1/2}$.\nWriting $z_j = x_j + i y_j$, the condition becomes $\\sum_j (x_j^2 + y_j^2) \u003c 1$, so $\\mathbb{B}^n$ is the Euclidean unit ball of $\\R^{2n}$.\n\n## Remarks\n\nIts boundary is the unit sphere $S^{2n-1} = \\set{z : \\abs{z} = 1}$.\nIt is a [[Reinhardt domain|WA0SE56FVH]]: if $z \\in \\mathbb{B}^n$ and $\\theta_j \\in \\R$, then $(e^{i\\theta_1} z_1, \\dots, e^{i\\theta_n} z_n)$ has the same norm, so it lies in $\\mathbb{B}^n$.\nEvery unitary map $U$ of $\\C^n$ preserves the norm, so it maps $\\mathbb{B}^n$ onto itself.\nFor $n = 1$ it is the unit disc, and for $n \\ge 2$ it is not biholomorphic to the unit [[polydisc|DJBWP7D3M8]], by [[Poincaré's theorem|1NF8T07H9X]].\n\n## Explanations\n\n### L1 human\n\nThe unit ball of $\\C^n$ is the set of points at distance less than $1$ from the origin.\nAs a set it is just the round ball of $\\R^{2n}$; what is new is that one studies the holomorphic functions on it.\nIn $\\C^2$, the point $(1/2, 1/2)$ is in the ball, since $1/4 + 1/4 \u003c 1$, while $(0.9, 0.9)$ is not.\nThe ball is very symmetric: its holomorphic automorphisms can move any point of it to the origin.\n",
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