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  "body": "\n## Summary\n\nA Fréchet space is a [[locally convex space|B6GY2KJMWW]] that is also an [[F-space|86BYEV9AX8]]: its topology comes from a complete translation-invariant metric.\nEquivalently, it is a complete Hausdorff space whose topology is generated by a countable family of seminorms.\nFréchet spaces generalize Banach spaces and include the main spaces of smooth and holomorphic functions.\nThe Hahn–Banach theorem, the open mapping theorem and the uniform boundedness principle all hold for them.\n\n## Explanations\n\n### L1 human\n\nA Banach space is complete for one norm.\nA Fréchet space is complete for a whole sequence of seminorms used together.\nThe space of smooth functions on $\\R$ is a typical case: one controls $f$, $f'$, $f''$, and so on, each on bounded intervals.\nNo single norm captures all this control, but countably many seminorms do.\nConvergence means convergence for each seminorm separately.\n",
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