In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Sazonov's theorem, named after
Vyacheslav Vasilievich Sazonov (), is a
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
.
It states that a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
between two
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
s is
''γ''-radonifying if it is a
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
\, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^ ...
. The result is also important in the study of
stochastic processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
and the
Malliavin calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...
, since results concerning
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s on infinite-dimensional spaces are of central importance in these fields. Sazonov's theorem also has a converse: if the map is not Hilbert–Schmidt, then it is not ''γ''-radonifying.
Statement of the theorem
Let ''G'' and ''H'' be two Hilbert spaces and let ''T'' : ''G'' → ''H'' be a
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
from ''G'' to ''H''. Recall that ''T'' is said to be ''γ''-radonifying if the
push forward of the
canonical Gaussian cylinder set measure on ''G'' is a ''bona fide''
measure on ''H''. Recall also that ''T'' is said to be a
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
\, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^ ...
if there is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
of ''G'' such that
:
Then Sazonov's theorem is that ''T'' is ''γ''-radonifying if it is a Hilbert–Schmidt operator.
The proof uses
Prokhorov's theorem
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered p ...
.
Remarks
The canonical Gaussian
cylinder set measure In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a Measure (mathematics), measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hil ...
on an infinite-dimensional Hilbert space can never be a ''bona fide'' measure; equivalently, the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on such a space cannot be ''γ''-radonifying.
See also
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References
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{{Functional analysis
Stochastic processes
Theorems in functional analysis
Theorems in measure theory