Classical Wiener space

Description

Let $T > 0$ and $n \ge 1$. The classical Wiener space is the set $$C_0([0, T]; \R^n) = \set{w \colon [0, T] \to \R^n : w \text{ continuous}, \ w(0) = 0},$$ with the norm $\norm{w} = \sup_{0 \le t \le T} \abs{w(t)}$ and the Wiener measure $\mu$. The 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 on $[0, T]$.

Check that $C_0([0, T]; \R^n)$ is a Banach space:

1. It is a vector space, and $\norm{\cdot}$ is a norm on it, since a continuous function on a compact interval is bounded.
2. A Cauchy sequence for $\norm{\cdot}$ converges uniformly to a function $w$; a uniform limit of continuous functions is continuous, and $w(0) = 0$.

It is also separable: piecewise linear paths with rational vertices are dense.