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  "body": "\n## Description\n\nLet $\\ell^2$ be the space of real sequences $x = (x_n)_{n \\ge 1}$ with $\\sum_n x_n^2 \u003c \\infty$ and the norm $\\norm{x}_2 = (\\sum_n x_n^2)^{1/2}$.\nThe *Hilbert cube* is the subset\n$$H = \\set{x \\in \\ell^2 : 0 \\le x_n \\le 1/n \\text{ for all } n \\ge 1}$$\nwith the metric induced by $\\norm{\\cdot}_2$.\n\n## Remarks\n\nThe map $(t_n)_{n \\ge 1} \\mapsto (t_n / n)_{n \\ge 1}$ is a [[homeomorphism|0QT00PB5G2]] from $[0, 1]^{\\N}$, with the [[product topology|0GBG8B49AA]], onto $H$.\nSo $H$ is [[compact|0TVJZYB6SH]], for example by [[Tychonoff's theorem|0JTRSJKCFB]].\nIt is also convex in $\\ell^2$.\n\nEvery [[separable|0Z2P41SVDP]] metrizable space is homeomorphic to a subspace of $H$.\nThis is the embedding behind the [[Urysohn metrization theorem|HBFHDH4EJX]].\n\nThe Hilbert cube has the [[fixed-point property|JQDHHFXBF3]]: every continuous map $H \\to H$ has a fixed point.\nThis follows from the Schauder fixed-point theorem, since $H$ is a compact convex subset of a Banach space.\n\n## Explanations\n\n### L1 human\n\nThe Hilbert cube is an infinite-dimensional cube whose sides get shorter and shorter: $1$, $1/2$, $1/3$, and so on.\nBecause the sides shrink, the cube stays compact, unlike the unit ball of an infinite-dimensional space.\nIt is big enough to contain a copy of every separable metric space, up to homeomorphism.\n",
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