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The [[Probability distribution|probability distribution]] of a [[Brownian motion|Brownian motion]]  $  \{ {B ( t ) } : {t \geq  0 } \} $
is a Gaussian measure (cf. also [[Constructive quantum field theory|Constructive quantum field theory]]) that can be supported by the space  $  C = C [ 0, \infty ) $
of continuous functions. For this reason,  $  C $
is also called the classical Wiener space. This notion can be generalized to a [[Banach space|Banach space]] on which a Gaussian measure can be introduced.

Advanced stochastic analysis can be carried out on a Wiener space. The analysis was initiated by P. Lévy and N. Wiener, and a systematic development was made by R.H. Cameron and W.T. Martin, I.E. Segal, L. Gross, K. Itô, and others.

Following [[#References|[a2]]], one can introduce the notion of an abstract Wiener space, which is a generalization of the classical Wiener space.

Let  $  H $
be a real separable [[Hilbert space|Hilbert space]] with norm  $  \| \cdot \| $. 
On  $  H $
one introduces the weak Gaussian distribution  $  \nu $
in such a way that on any finite-, say  $  n $-, 
dimensional subspace  $  K $
of  $  H $
the restriction of  $  \nu $
to  $  K $
is the  $  n $-
dimensional standard Gaussian distribution. In fact,  $  \nu $
may be called the weak white noise measure. A semi-norm (or norm)  $  \| \cdot \| _ {1} $
on  $  H $
is called a measurable norm if for any positive  $  \epsilon $
there exists a finite-dimensional projection operator  $  P _ {0} $
such that for any finite-dimensional projection operator  $  P $
orthogonal to  $  P _ {0} $
the inequality  $  \nu \{ x : {\| {Px } \| _ {1} > \epsilon } \} < \epsilon $
holds.

Now, let  $  \| x \| _ {1} $
be a measurable norm on  $  H $
and let  $  B $
be the completion of  $  H $
with respect to this norm (cf. [[Complete space|Complete space]]). Then  $  B $
is a Banach space. Let  $  {\mathcal R} $
be the  $  \sigma $-
ring generated by the cylinder subsets of  $  B $(
cf. [[Cylinder set|Cylinder set]]). For a cylinder set measure  $  \mu $
on  $  {\mathcal R} $
induced by the Gaussian distribution on  $  H $, 
the measure  $  \mu $
is countably additive on  $  {\mathcal R} $. 
Therefore, taking the  $  \sigma $-
field  $  {\mathcal B} $
generated by  $  {\mathcal R} $, 
a [[Measure space|measure space]]  $  ( H, {\mathcal B}, \mu ) $
is obtained.

The classical definition of an abstract Wiener space is given as follows. First, take a Hilbert space  $  H $
with norm  $  \| \cdot \| $
and take a measurable norm  $  \| \cdot \| _ {1} $, 
to obtain a Banach space  $  B $. 
The injection mapping from  $  H $
into  $  B $
is denoted by  $  i $. 
Then the triple  $  ( i,H,B ) $
is called an abstract Wiener space. This means that a weak measure on  $  H $
can be extended to a completely additive measure supported by  $  B $. 
A stochastic analysis can be developed for this latter measure (see [[#References|[a4]]]).

One of the developments of the notion of an abstract Wiener space is that of a [[Rigged Hilbert space|rigged Hilbert space]], due to I.M. Gel'fand and N.Ya. Vilenkin (see [[#References|[a1]]]). Let  $  H $
be a real Hilbert space and let  $  \Phi $
be a countably Hilbert [[Nuclear space|nuclear space]] that is continuously imbedded in  $  H $. 
The dual space  $  \Phi  ^ {*} $
of  $  \Phi $
gives rise to the rigged Hilbert space

$$ 
\Phi \subset  H \subset  \Phi  ^ {*} .
$$

Given a characteristic functional  $  C ( \xi ) $, 
$  \xi \in \Phi $, 
that is,  $  C ( \xi ) $
is continuous in  $  \xi $, 
positive definite and  $  C ( 0 ) = 1 $, 
there exists a countably additive probability measure  $  \mu $
in  $  \Phi  ^ {*} $
such that

$$ 
C ( \xi ) = \int\limits _ {\Phi  ^ {*} } { { \mathop{\rm exp} } [ i \left \langle  {x, \xi } \right \rangle ] }  {d \mu ( x ) } .
$$

A typical example of a rigged Hilbert space is the triple consisting of the Hilbert space  $  L _ {2} ( \mathbf R ) $, 
the Schwartz space  $  S $
and the space  $  S  ^ {*} $
of tempered distributions (cf. [[Generalized function|Generalized function]]). [[White noise|White noise]] is also an important example of  $  \mu $
on  $  S  ^ {*} $; 
it has characteristic functional  $  C ( \xi ) = { \mathop{\rm exp} } [ - { {\| \xi \|  ^ {2} } / 2 } ] $. 
The analysis on the function space with the white noise measure is well developed (see [[#References|[a3]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,   N.Ya. Vilenkin,   "Generalized functions 4: applications of harmonic analysis" , Acad. Press  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gross,   "Abstract Wiener spaces" , ''Proc. 5th Berkeley Symp. Math. Stat. Probab.'' , '''2'''  (1965)  pp. 31–42</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Hida,   H.H. Kuo,   J. Potthoff,   L. Streit,   "White noise. An infinite dimensional calculus" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.H. Kuo,   "Gaussian measures in Banach spaces" , ''Lecture Notes in Mathematics'' , '''463''' , Springer  (1975)</TD></TR></table>