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  "body": "\n## Summary\n\nA locally convex space is a topological vector space in which every neighbourhood of $0$ contains a convex neighbourhood of $0$.\nEquivalently, its topology is [[generated by a family of seminorms|ZVFXQ7S6FY]].\nNormed spaces, [[Fréchet spaces|3GAA614673]], spaces of smooth functions and spaces of distributions are all locally convex.\nLocal convexity is what makes the Hahn–Banach theorem work, so these spaces have many continuous linear forms and a rich duality theory.\n\n## Explanations\n\n### L1 human\n\nIn a normed space the small balls around $0$ are convex: the segment between two points of a ball stays in the ball.\nA locally convex space keeps this feature, but the small neighbourhoods need not come from one norm.\nThey may come from many seminorms used together.\nFor example, the space of all real sequences, with convergence meaning convergence of each coordinate, is locally convex but has no norm giving that convergence.\nSome natural spaces fail local convexity, such as $L^p([0, 1])$ for $0 \u003c p \u003c 1$.\n",
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