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  "body": "\n## Summary\n\nThe completion of a normed space $X$ is a [[Banach space|06GZ956657]] that contains a copy of $X$ as a dense subspace, with the same norm.\nIt exists and is unique up to isometric isomorphism.\nIt lets one prove results for Banach spaces and apply them to incomplete spaces.\nMany important spaces, such as $L^p$ spaces and Sobolev spaces, can be built as completions of spaces of nice functions.\n\n## Explanations\n\n### L1 human\n\nA normed space may have holes: Cauchy sequences that do not converge to anything inside it.\nThe completion fills these holes and changes nothing else, much as the real numbers fill the gaps in the rationals.\nFor example, finitely supported sequences with the norm $\\left(\\sum_k \\abs{x_k}^2\\right)^{1/2}$ do not form a complete space.\nTheir completion is the space $\\ell^2$ of all square-summable sequences.\n",
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