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  "body": "\n## Definition\n\nLet $H$ be a real or complex [[Hilbert space|HM2HHE3J51]].\nAn *orthonormal basis* of $H$ is an [[orthonormal family|0RW9TNNT87]] $(e_i)_{i \\in I}$ in $H$ that is maximal: no nonzero vector of $H$ is orthogonal to all $e_i$.\nAll orthonormal bases of $H$ have the same [[cardinality|0G0ASKNA6Z]].\nThe *Hilbert dimension* of $H$ is this cardinality $\\abs{I}$.\n\n## Remarks\n\nExample: in $\\ell^2$, the unit vectors $e_n$, $n \\ge 1$, form an orthonormal basis.\nThey are orthonormal, and if $\\ip{x}{e_n} = x_n = 0$ for all $n$ then $x = 0$.\nSo the Hilbert dimension of $\\ell^2$ is $\\aleph_0$, the cardinality of $\\N$.\n\nNon-example: the family $(e_n)_{n \\ge 2}$ in $\\ell^2$ is orthonormal but not maximal, since $e_1$ is orthogonal to all of it.\nSo it is not an orthonormal basis, though its cardinality is also $\\aleph_0$.\n\nFor every orthonormal basis, each $x \\in H$ is the sum $x = \\sum_{i \\in I} \\ip{x}{e_i} e_i$, with only countably many nonzero terms.\nFor finite-dimensional $H$, the Hilbert dimension is the usual dimension.\nFor $H = L^2([0, 2\\pi])$, the functions $e^{inx} / \\sqrt{2\\pi}$, $n \\in \\Z$, form an orthonormal basis, so $L^2([0, 2\\pi])$ and $\\ell^2$ are isometrically isomorphic.\n",
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