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  "body": "\n## Summary\n\nAn F-space is a topological vector space whose topology comes from a complete translation-invariant metric.\nF-spaces are the natural setting of the Baire category arguments of functional analysis: the open mapping theorem and the closed graph theorem hold for F-spaces.\nEvery Banach space and every [[Fréchet space|3GAA614673]] is an F-space, but F-spaces need not be locally convex.\n\n## Explanations\n\n### L1 human\n\nAn F-space is a vector space with a notion of distance that behaves well under shifting and has no missing limit points.\nShifting two vectors by the same amount does not change their distance.\nEvery Cauchy sequence converges.\nThe space $L^p([0, 1])$ with $0 \u003c p \u003c 1$ and distance $d(f, g) = \\int_0^1 \\abs{f - g}^p$ is an F-space.\nIts only open convex set containing $0$ is the whole space, so it is not locally convex, and in particular it is not a Banach space.\n",
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