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  "body": "\n## Summary\n\nA subset of a topological vector space is von Neumann bounded if a large enough scaling shrinks it into any given neighbourhood of $0$.\nIt replaces the notion of a [[bounded set|09QSZVVM26]] of a metric space.\nThat notion does not suit topological vector spaces, since a metric may make the whole space bounded.\nIn a normed space the von Neumann bounded sets are the sets of finite norm diameter.\nBounded sets are used to define the [[strong dual space|RG52DZ1A8Y]], [[bornological spaces|4R6SQD60FN]] and [[Montel spaces|NWMYW2WK9S]].\n\n## Explanations\n\n### L1 human\n\nPick any small neighbourhood of the origin, however tiny.\nA set is bounded if, after you divide it by a large enough number, it fits inside that neighbourhood.\nIn $\\R^2$ this is the usual idea: a set is bounded if it fits inside some disc.\nA straight line through the origin is not bounded, because no rescaling of it fits in a small disc.\nIn a space of functions with seminorms $p_n$, a set is bounded when each $p_n$ is bounded on it, possibly with a different bound for each $n$.\n",
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