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  "body": "\n## Description\n\nLet $T \u003e 0$ and $n \\ge 1$.\nThe classical Wiener space is the set\n$$C_0([0, T]; \\R^n) = \\set{w \\colon [0, T] \\to \\R^n : w \\text{ continuous}, \\ w(0) = 0},$$\nwith the norm $\\norm{w} = \\sup_{0 \\le t \\le T} \\abs{w(t)}$ and the Wiener measure $\\mu$.\nThe Wiener measure is the unique Borel probability measure on this space under which the coordinate process $W_t(w) = w(t)$ is an $n$-dimensional [[Wiener process|552AWX3QXN]] on $[0, T]$.\n\nCheck that $C_0([0, T]; \\R^n)$ is a [[Banach space|06GZ956657]]:\n\n1. It is a vector space, and $\\norm{\\cdot}$ is a norm on it, since a continuous function on a compact interval is bounded.\n2. A Cauchy sequence for $\\norm{\\cdot}$ converges uniformly to a function $w$; a uniform limit of continuous functions is continuous, and $w(0) = 0$.\n\nIt is also separable: piecewise linear paths with rational vertices are dense.\n\n## Remarks\n\nThe Wiener measure is the law of Brownian motion, seen as a random element of the path space.\nThe Cameron–Martin space is the subspace of absolutely continuous paths $h$ with $h(0) = 0$ and $\\int_0^T \\abs{h'(t)}^2 \\, dt \u003c \\infty$.\nTranslations of $\\mu$ by elements of the Cameron–Martin space give measures equivalent to $\\mu$, but the Cameron–Martin space itself has $\\mu$-measure zero.\nThe space is the standard setting of [[Malliavin calculus|1EC7VPHFAY]].\n\n## Explanations\n\n### L1 human\n\nThe classical Wiener space is the set of all possible paths of a Brownian particle that starts at the origin.\nEach point of the space is a whole continuous path, and two paths are close when they stay close at every time.\nThe Wiener measure tells how likely each set of paths is, so that \"pick a random path\" means \"run a Brownian motion\".\n",
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