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  "body": "\n## Summary\n\nA topological vector space is a [[vector space|000AYPA9AJ]] over $\\R$ or $\\C$ with a topology in which addition and scalar multiplication are continuous.\nIt is the common frame of functional analysis: normed spaces, Banach spaces, Hilbert spaces and locally convex spaces are all examples.\nUnder addition it is a commutative [[topological group|92700QAB5Y]], so its topology is determined by the neighbourhoods of $0$.\nContinuity of linear maps, convergence of series and duality are all studied in this setting.\n\n## Explanations\n\n### L1 human\n\nA topological vector space is a space of vectors where you can also say when vectors are close to each other.\nAdding vectors and stretching them by numbers must respect this closeness.\nThe plane $\\R^2$ with its usual distance is the simplest example.\nSpaces of functions, such as all continuous functions on $[0, 1]$ with the maximum distance, are the examples that matter most in analysis.\n",
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