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  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] whose inner product $\\ip{\\cdot}{\\cdot}$ is linear in the first argument.\nLet $\\Phi$ be a subspace of $H$ with a locally convex topology such that these conditions hold:\n\n1. The inclusion $\\Phi \\to H$ is continuous.\n2. $\\Phi$ is dense in $H$.\n\nLet $\\Phi^\\times$ be the space of continuous conjugate-linear maps $\\Phi \\to \\C$.\nEach $h \\in H$ gives the element $\\varphi \\mapsto \\ip{h}{\\varphi}$ of $\\Phi^\\times$, and the resulting map $H \\to \\Phi^\\times$ is linear and injective.\nThe chain $\\Phi \\subset H \\subset \\Phi^\\times$ is a *rigged Hilbert space*, or *Gelfand triple*.\nIn the theory of generalized eigenvectors one also asks $\\Phi$ to be a [[nuclear space|XC6XPMC2FQ]].\n\n## Remarks\n\nInjectivity of $H \\to \\Phi^\\times$ comes from condition 2: if $\\ip{h}{\\varphi} = 0$ for all $\\varphi$ in a dense subspace, then $h = 0$.\nMany authors use the [[continuous dual|4A82S5FVRT]] $\\Phi'$ of linear forms instead of $\\Phi^\\times$; then the map $H \\to \\Phi'$ is conjugate-linear.\n\nExample: $\\Phi = \\mathcal{S}(\\R)$, the Schwartz space with its Fréchet topology, $H = L^2(\\R)$, and $\\Phi^\\times$ the conjugate-linear tempered distributions.\nThe inclusion is continuous because $\\norm{\\varphi}_{L^2} \\le \\sqrt{\\pi}\\, \\sup_x \\abs{(1 + x^2)^{1/2}\\varphi(x)}$, and $\\mathcal{S}(\\R)$ is dense in $L^2(\\R)$.\nThe delta $\\varphi \\mapsto \\overline{\\varphi(a)}$ lies in $\\Phi^\\times$ but not in $H$.\n\nNon-example: let $\\Phi$ be the Schwartz functions that vanish on $[0,1]$.\nIts closure in $L^2(\\R)$ consists of functions that vanish almost everywhere on $[0,1]$, so condition 2 fails.\nThen every $h \\in L^2(\\R)$ supported in $[0,1]$ gives the zero functional on $\\Phi$, and $H \\to \\Phi^\\times$ is not injective.\n",
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