In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually "n"-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.
Definition.
Consider "E" ⊆ R"n" 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",
formula_1 as formula_2
In almost all applications, one takes "E" = [0, "T" ] or [0, +∞) and "M" = R"n" for some "n" in N. For brevity, write "C" for "C"([0, "T" ]; R"n"); this is a vector space. Write "C"0 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 "C"0 as "classical Wiener space".
For a stochastic process formula_3 and the space formula_4 of all functions from formula_5 to formula_6, one looks at the map formula_7. One can then define the "coordinate maps" or "canonical versions" formula_8 defined by formula_9. The formula_10 form another process. The "Wiener measure" is then the unique measure on formula_11 such that the coordinate process is a Brownian motion.
Properties of classical Wiener space.
Uniform topology.
The vector space "C" can be equipped with the uniform norm
formula_12
turning it into a normed vector space (in fact a Banach space since formula_13 is compact). This norm induces a metric on "C" in the usual way: formula_14. The topology generated by the open sets in this metric is the topology of uniform convergence on [0, "T" ], or the uniform topology.
Thinking of the domain [0, "T" ] as "time" and the range R"n" as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space slightly" and get the graph of "f" to lie on top of the graph of "g", while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.
If one looks at the more general domain formula_15 with 
formula_16
then the Wiener space is no longer a Banach space, however it can be made into one if the Wiener space is defined under the additional constraint
formula_17
Separability and completeness.
With respect to the uniform metric, "C" is both a separable and a complete space:
Since it is both separable and complete, "C" is a Polish space.
Tightness in classical Wiener space.
Recall that the modulus of continuity for a function "f" : [0, "T" ] → R"n" is defined by
formula_18
This definition makes sense even if "f" is not continuous, and it can be shown that "f" is continuous if and only if its modulus of continuity tends to zero as δ → 0:
formula_19.
By an application of the Arzelà-Ascoli theorem, one can show that a sequence formula_20 of probability measures on classical Wiener space "C" is tight if and only if both the following conditions are met:
formula_21 and
formula_22 for all ε ; 0.
Classical Wiener measure.
There is a "standard" measure on "C"0, known as classical Wiener measure (or simply Wiener measure). Wiener measure has (at least) two equivalent characterizations:
If one defines Brownian motion to be a Markov stochastic process "B" : [0, "T" ] × Ω → R"n", starting at the origin, with almost surely continuous paths and independent increments
formula_23
then classical Wiener measure γ is the law of the process "B".
Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure γ is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to "C"0.
Classical Wiener measure is a Gaussian measure: in particular, it is a strictly positive probability measure.
Given classical Wiener measure γ on "C"0, the product measure γ&amp;hairsp;"n" × γ is a probability measure on "C", where γ&amp;hairsp;"n" denotes the standard Gaussian measure on R"n".
Subspaces of the Wiener space.
Let formula_24 be a Hilbert space that is continously embbeded and let formula_25 be the Wiener measure then formula_26. This was proven in 1973 by Smolyanov and Uglanov and in the same year independently by Guerquin. However, there exists a Hilbert space formula_24 with weaker topology such that formula_28 which was proven in 1993 by Uglanov.