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  "body": "\n## Summary\n\nA Fréchet manifold is a space that looks locally like an open subset of a Fréchet space, with smooth changes of coordinates.\nIt is the natural setting for spaces of smooth maps, such as the smooth loops in a manifold or the diffeomorphisms of a compact manifold.\nCalculus on Fréchet manifolds is harder than on Banach manifolds, because the inverse function theorem fails in general Fréchet spaces.\nThe Nash–Moser theorem gives a replacement under extra hypotheses.\n\n## Explanations\n\n### L1 human\n\nSpaces of smooth functions are not Banach spaces, because no single norm controls all derivatives at once.\nThey are Fréchet spaces, controlled by a sequence of norms.\nA Fréchet manifold is built from patches that look like pieces of such spaces.\nFor example, the space of all smooth loops in a sphere is a Fréchet manifold.\n",
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