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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$ with its usual topology.\nA *topological vector space* over $\\mathbb{K}$ is a [[vector space|000AYPA9AJ]] $E$ over $\\mathbb{K}$ together with a [[topology|0CQHFPGJ45]] on $E$ such that:\n\n1. the addition $E \\times E \\to E$, $(x, y) \\mapsto x + y$, is [[continuous|0VGFHTM75Y]];\n2. the scalar multiplication $\\mathbb{K} \\times E \\to E$, $(\\lambda, x) \\mapsto \\lambda x$, is continuous.\n\nBoth products carry the [[product topology|0GBG8B49AA]].\n\n## Remarks\n\nExample: $\\R^n$ with the Euclidean topology.\nAddition and scalar multiplication are given by polynomials in the coordinates, so both are continuous.\nNon-example: $\\R$ as a vector space over $\\R$ with the discrete topology.\nAddition is continuous, but scalar multiplication fails clause 2.\nIf it were continuous, then the map $\\lambda \\mapsto \\lambda \\cdot 1$ from $\\mathbb{K}$ to $E$ would be continuous.\nThen the preimage $\\set{1}$ of the open set $\\set{1}$ would be open in $\\mathbb{K}$, which is false.\nMany authors also require the topology to be Hausdorff.\nThe definition extends to vector spaces over any topological field.\n",
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