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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a topological vector space over $\\mathbb{K}$.\nA subset $B \\subseteq E$ is *von Neumann bounded*, or *bounded*, if the following holds.\nFor every [[neighbourhood|0ZCY0Z80B6]] $U$ of $0$ in $E$ there is $s \u003e 0$ such that $B \\subseteq t U$ for every $t \\in \\mathbb{K}$ with $\\abs{t} \u003e s$.\n\n## Remarks\n\nSince the balanced neighbourhoods of $0$ form a neighbourhood base, an equivalent condition is: for every neighbourhood $U$ of $0$ there is $t \u003e 0$ with $B \\subseteq t U$.\nIf the topology of $E$ is generated by a family $\\mathcal{P}$ of seminorms, then $B$ is bounded if and only if $\\sup_{x \\in B} p(x) \u003c \\infty$ for every $p \\in \\mathcal{P}$.\nExample: in a normed space, the closed unit ball $B$ is bounded; a neighbourhood $U$ of $0$ contains a ball of some radius $\\varepsilon \u003e 0$, and then $B \\subseteq tU$ for all $\\abs{t} \u003e 1 / \\varepsilon$.\nNon-example: in $\\R^2$ with the Euclidean norm, the line $L = \\R \\times \\set{0}$ is not bounded; for $U$ the open unit disc, $tU$ is the disc of radius $\\abs{t}$, which never contains $L$.\nSubsets of bounded sets, finite unions of bounded sets, closures of bounded sets and compact sets are bounded.\n",
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