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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $U$ be a subspace of $V$.\nThen $U$ has a [[complement|0M83V17KHX]] in $V$: there is a subspace $W$ of $V$ with $V = U \\oplus W$.\n\n## Remarks\n\nProof: extend a basis $B_U$ of $U$ to a basis $B$ of $V$, and let $W$ be the span of $B \\setminus B_U$.\nFor infinite-dimensional $V$, this uses the extension of independent sets to bases, and hence Zorn's lemma.\nThe statement fails for modules over rings: the submodule $2\\Z$ of $\\Z$ has no complement.\nFor closed subspaces of topological vector spaces, a closed complement need not exist.\n\n## Explanations\n\n### L1 human\n\nEvery subspace has a partner subspace that fills up the rest of the space.\nWe find it by completing a basis of the subspace and taking the span of the new vectors.\nThe partner is not unique, but its dimension is fixed.\n",
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