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  "body": "\n## Definition\n\nLet $H$ be a real separable [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$ and norm $\\norm{\\cdot}_H$.\nLet $E$ be a real [[separable|0Z2P41SVDP]] [[Banach space|06GZ956657]] with dual space $E^*$.\nLet $i \\colon H \\to E$ be an injective [[continuous linear map|03TM135G0N]] with dense image.\nFor $\\ell \\in E^*$, let $i^* \\ell \\in H$ be the vector with $\\ip{i^* \\ell}{h} = \\ell(i h)$ for all $h \\in H$.\nThe triple $(i, H, E)$ is an *abstract Wiener space* if there is a Borel [[probability measure|059TJN0NZ1]] $\\mu$ on $E$ with the following property.\nFor every $\\ell \\in E^*$, the [[pushforward|3N66TM3TSQ]] $\\ell_* \\mu$ is the centered [[normal distribution|05AQE0QR0D]] with variance $\\norm{i^* \\ell}_H^2$.\n\n## Remarks\n\nThe measure $\\mu$ is unique, and it is called the abstract Wiener measure; it is a centered [[Gaussian measure|P0DV2TCH1N]].\nEquivalently, the image under $i$ of the canonical Gaussian [[cylinder set measure|G95YDNFXQN]] of $H$ extends to a countably additive measure on $E$.\nConversely, every centered Gaussian measure on a separable Banach space arises in this way, with $H$ its Cameron–Martin space.\n\nExample: let $E = C_0([0, 1])$ be the [[classical Wiener space|5Z00K7K1HY]] with the sup norm.\nLet $H$ be the space of absolutely continuous $h \\in E$ with $h' \\in L^2$, with $\\ip{h}{k} = \\int_0^1 h' k' \\, dt$, and let $i$ be the inclusion.\nWith $\\mu$ the Wiener measure, take $\\ell(\\omega) = \\omega(t)$.\nThen $i^* \\ell$ is $s \\mapsto \\min(s, t)$, whose norm squared is $t$, and indeed $\\omega(t)$ has law $N(0, t)$.\n\nNon-example: let $E = H$ be infinite-dimensional and $i$ the identity.\nFor an orthonormal sequence $(e_k)$, the readings $\\ip{x}{e_k}$ would be independent standard normal variables under $\\mu$.\nThen $\\sum_k \\ip{x}{e_k}^2 = \\infty$ almost surely, which is impossible for $x \\in H$, so no such $\\mu$ exists.\n\n## Explanations\n\n### L1 human\n\nThe Hilbert space fixes the shape of the Gaussian, and the Banach space is large enough to carry the actual measure.\nWiener measure on continuous paths is the model case.\n",
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