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  "body": "\n## Summary\n\nThe tensor product of two Hilbert spaces is the completion of their algebraic tensor product.\nIts inner product multiplies the inner products of the factors.\nIt is again a Hilbert space, and products of orthonormal basis vectors form an orthonormal basis of it.\nIn quantum mechanics it describes a system made of two subsystems.\nIn analysis, $L^2(X) \\mathbin{\\widehat{\\otimes}} L^2(Y)$ is $L^2(X \\times Y)$.\n\n## Explanations\n\n### L1 human\n\nIf one Hilbert space describes functions of $x$ and another describes functions of $y$, the tensor product describes functions of the pair $(x, y)$.\nThe simplest elements are products $f(x) g(y)$, and general elements are limits of sums of such products.\nFor example, $\\C^2 \\otimes \\C^3$ is $\\C^6$, with the six products $e_i \\otimes f_j$ of basis vectors as an orthonormal basis.\n",
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