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  "body": "\n## Statement\n\nLet $E$ be an infinite-dimensional [[separable|0Z2P41SVDP]] real [[Fréchet space|3GAA614673]], for example an infinite-dimensional separable [[Banach space|06GZ956657]].\nThen $E$ is [[homeomorphic|0QT00PB5G2]] to the countable product $\\R^{\\N}$ with the product topology.\nIn particular, any two such spaces are homeomorphic, and each of them is homeomorphic to the separable [[Hilbert space|HM2HHE3J51]] $\\ell^2$.\n\n## Remarks\n\nKadec proved that all infinite-dimensional separable Banach spaces are homeomorphic, and Anderson proved that $\\R^{\\N}$ is homeomorphic to $\\ell^2$.\nThe homeomorphisms are far from linear: $\\ell^1$ and $\\ell^2$ are homeomorphic but not isomorphic as Banach spaces.\nBoth hypotheses are needed.\nIn finite dimension, $\\R^m$ and $\\R^n$ are not homeomorphic for $m \\ne n$, by invariance of domain, and neither is homeomorphic to $\\R^{\\N}$.\nWithout separability it fails: $\\ell^\\infty$ is not separable, so it is not homeomorphic to $\\ell^2$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that, as far as topology alone is concerned, there is only one separable infinite-dimensional Fréchet space.\nNorms, inner products and linear structure distinguish many such spaces, but if one only looks at open sets they all look like the product of countably many lines.\nSo topology cannot tell $\\ell^1$ from $\\ell^2$, even though linear algebra can.\n\n### L2 human\n\nThe result is the starting point of infinite-dimensional topology.\nSince all separable infinite-dimensional Fréchet spaces are homeomorphic, manifolds modelled on any of them can be compared purely topologically.\nFor Hilbert manifolds the smooth counterpart is the open embedding theorem.\nIt places every separable Hilbert manifold modelled on an infinite-dimensional separable Hilbert space inside its model space as an open set.\n",
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