Classical Wiener space

In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a sub-interval of the real line), taking values in a metric space (usually n-dimensional Euclidean space). Classical Wiener space is useful in the study of whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.

Consider E ⊆ Rⁿ and a metric space (M, d). The classical Wiener space C(E; M) is the space of all continuous functions f : E → M. I.e. for every fixed t in E,

In almost all applications, one takes E = [0, T] or [0, +∞) and M = Rⁿ for some n in N. For brevity, write C for C([0, T]; Rⁿ); this is a vector space. Write C₀ for the linear subspace consisting only of those functions that take the value zero at the infimum of the set E. Many authors refer to C₀ as "classical Wiener space".

The vector space C can be equipped with the uniform norm

turning it into a normed vector space (in fact a Banach space). This norm induces a metric on C in the usual way: ${\displaystyle d(f,g):=\|f-g\|}$. The topology generated by the open sets in this metric is the topology of uniform convergence on [0, T], or the .

...
Wikipedia