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  "body": "\n## Summary\n\nEvery topological vector space is, under addition, a commutative topological group.\nThis additive structure is the source of many basic facts about topological vector spaces.\nTranslations are homeomorphisms, so the whole topology is fixed by the neighbourhoods of $0$.\nIt also gives each topological vector space a canonical uniform structure, so Cauchy nets and completeness make sense without a metric.\n\n## Explanations\n\n### L1 human\n\nIn a topological vector space you may add vectors, and adding is a continuous operation.\nSo, if you forget scalar multiplication, what is left is a group with a topology that fits the group law.\nShifting the whole space by a fixed vector moves open sets to open sets.\nHence, to know what \"near $x$\" means, it is enough to know what \"near $0$\" means and then shift.\nFor example, in $\\R^2$ the open discs around a point $x$ are exactly the open discs around $0$ moved by $x$.\n",
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