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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $H_1$ and $H_2$ be [[Hilbert spaces|HM2HHE3J51]] over $\\mathbb{K}$.\nLet $H_1 \\otimes H_2$ be their algebraic [[tensor product|0BF2DE2SYK]] over $\\mathbb{K}$.\nThere is exactly one inner product on $H_1 \\otimes H_2$ such that\n$$\\ip{x_1 \\otimes x_2}{y_1 \\otimes y_2} = \\ip{x_1}{y_1} \\ip{x_2}{y_2}$$\nfor all $x_1, y_1 \\in H_1$ and $x_2, y_2 \\in H_2$.\nThe *tensor product of Hilbert spaces* $H_1 \\mathbin{\\widehat{\\otimes}} H_2$ is the completion of $H_1 \\otimes H_2$ for this inner product.\n\n## Remarks\n\nIf $(e_i)_{i \\in I}$ and $(f_j)_{j \\in J}$ are [[orthonormal bases|0S9GGTR0EX]] of $H_1$ and $H_2$, then $(e_i \\otimes f_j)_{(i, j) \\in I \\times J}$ is an orthonormal basis of $H_1 \\mathbin{\\widehat{\\otimes}} H_2$.\n\nExample: $\\ell^2(I) \\mathbin{\\widehat{\\otimes}} \\ell^2(J)$ is unitarily equivalent to $\\ell^2(I \\times J)$, by $e_i \\otimes f_j \\mapsto e_{(i, j)}$.\nBoth sides have orthonormal bases indexed by $I \\times J$, and the map sends one onto the other.\nSimilarly, for $\\sigma$-finite measure spaces, $L^2(X) \\mathbin{\\widehat{\\otimes}} L^2(Y)$ is $L^2(X \\times Y)$, with $f \\otimes g$ sent to $(x, y) \\mapsto f(x) g(y)$.\n\nNon-example: if $H_1$ and $H_2$ are infinite-dimensional, the algebraic tensor product $H_1 \\otimes H_2$ with the inner product above is not complete.\nFor instance, $\\sum_{n \\ge 1} 2^{-n} e_n \\otimes f_n$ converges in the completion but is not a finite sum of elementary tensors.\nIn finite dimension no completion is needed, and $H_1 \\mathbin{\\widehat{\\otimes}} H_2 = H_1 \\otimes H_2$.\n\nOther completions of $H_1 \\otimes H_2$, for instance the projective tensor product, are Banach spaces with different norms.\n",
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