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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$.\nThen $V$ has an [[orthonormal basis|0S9GGTR0EX]].\nMoreover, each orthonormal family in $V$ extends to an orthonormal basis of $V$.\n\n## Remarks\n\nThe proof applies the [[Gram–Schmidt process|0VT6P0GNQR]] to any basis of $V$.\nFor the extension, first extend the orthonormal family to a basis, and then apply Gram–Schmidt; it keeps the first vectors.\nHence each finite-dimensional real inner product space is isometric to $\\R^n$ with the dot product.\n\n## Explanations\n\n### L1 human\n\nEvery finite-dimensional inner product space has a basis of perpendicular unit vectors.\nWe construct one with the Gram–Schmidt process.\nSo every such space looks like $\\R^n$ or $\\C^n$ with the standard inner product.\n",
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