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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a topological vector space over $\\mathbb{K}$.\nThe *topological group structure* of $E$ is the additive group $(E, +)$ together with the topology of $E$.\nIt is a commutative topological group: the maps $(x, y) \\mapsto x + y$ from $E \\times E$ to $E$ and $x \\mapsto -x$ from $E$ to $E$ are continuous.\n\n## Remarks\n\nThe map $x \\mapsto -x$ is continuous because it is scalar multiplication by $-1$.\nFor each $a \\in E$ the translation $x \\mapsto x + a$ is a homeomorphism of $E$, with inverse $x \\mapsto x - a$.\nSo a set $U$ is a neighbourhood of $a$ if and only if $U - a$ is a neighbourhood of $0$.\nExample: $\\R^n$ with its Euclidean topology is a commutative topological group under addition, since addition and negation are continuous functions of the coordinates.\nNon-example: give $\\R$ the topology whose open sets are $\\emptyset$, $\\R$ and the intervals $(a, \\infty)$.\nThis is not a topological group, because $x \\mapsto -x$ is not continuous: the preimage of $(0, \\infty)$ is $(-\\infty, 0)$, which is not open.\nThe topological group structure gives a uniform structure on $E$; a net $(x_i)$ is Cauchy when $x_i - x_j$ tends to $0$.\n",
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