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  "body": "\n## Statement\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a [[Hausdorff|0S1FJ8VEX9]] topological vector space over $\\mathbb{K}$.\nThen some [[norm|0V2WP9B1MF]] on $E$ induces the topology of $E$ if and only if $0$ has a [[convex|02AS45T3H9]] [[von Neumann bounded|3BAJ8ZTVPX]] [[neighbourhood|0ZCY0Z80B6]] in $E$.\n\n## Remarks\n\nIf the topology comes from a norm, the open unit ball is a convex bounded neighbourhood of $0$.\nConversely, a convex bounded neighbourhood of $0$ contains a convex balanced open neighbourhood $U$ of $0$, which is still bounded.\nIts [[Minkowski functional|PAX9SHDHDW]] $p_U$ is a seminorm, and boundedness of $U$ shows that the sets $t U$, $t \u003e 0$, form a neighbourhood base at $0$, so $p_U$ induces the topology.\nThe Hausdorff property then forces $p_U(x) \u003e 0$ for $x \\ne 0$, so $p_U$ is a norm.\nExample: the Fréchet space $C(\\R)$ with the seminorms $p_n(f) = \\sup_{\\abs{t} \\le n} \\abs{f(t)}$ is not normable, because no neighbourhood of $0$ is bounded.\nThe space $L^p([0, 1])$ with $0 \u003c p \u003c 1$ is not normable either: it has bounded neighbourhoods of $0$ but no convex ones other than the whole space.\nThe Hausdorff hypothesis is needed: $\\R$ with the indiscrete topology is a topological vector space.\nIn it, $\\R$ itself is a convex bounded neighbourhood of $0$, but no norm induces this topology.\n\n## Explanations\n\n### L1 human\n\nThe theorem tells exactly when the topology of a topological vector space comes from a single norm.\nTwo features are needed near the origin: some neighbourhood must be convex, and it must also be small, in the sense of being bounded.\nA norm ball has both features, and the theorem says that a set with both features can always be turned into a norm ball.\n\n### L2 human\n\nLocal convexity means that the topology is generated by seminorms.\nSo the criterion says that a locally convex space is normable precisely when one seminorm suffices, up to equivalence.\nIt shows that spaces such as $C^\\infty(\\Omega)$, $H(\\Omega)$ or the Schwartz space are not normable.\nIn each of them a bounded neighbourhood of $0$ would be relatively compact, which happens only in finite dimension.\nA topological vector space with a bounded neighbourhood of $0$ is called locally bounded, so the criterion reads: normable means Hausdorff, locally convex and locally bounded.\nThe construction of the norm is the same Minkowski functional argument that links convex sets and seminorms in the theory of [[locally convex spaces|B6GY2KJMWW]].\n",
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