In
mathematics, more specifically 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 (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, a positive linear functional on an
ordered vector space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition
Given a vector space ''X'' over the real numbers R and a pr ...
is a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
on
so that for all
positive element In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \la ...
s
that is
it holds that
In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as
Riesz–Markov–Kakutani representation theorem
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuo ...
.
When
is a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
vector space, it is assumed that for all
is real. As in the case when
is a
C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace
and the partial order does not extend to all of
in which case the positive elements of
are the positive elements of
by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any
equal to
for some
to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such
This property is exploited in the
GNS construction
GNS may refer to:
Places
* Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia
* Gainesville station (Georgia), an Amtrak station in Georgia, United States
Companies and organizations
* Gesellschaft für Nuklear-Service, a German nuclear ...
to relate positive linear functionals on a C*-algebra to
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
s.
Sufficient conditions for continuity of all positive linear functionals
There is a comparatively large class of
ordered topological vector spaces on which every positive linear form is necessarily continuous.
This includes all
topological vector lattices that are
sequentially complete
In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in .
is called sequentially complete if i ...
.
Theorem Let
be an
Ordered topological vector space with
positive cone and let
denote the family of all bounded subsets of
Then each of the following conditions is sufficient to guarantee that every positive linear functional on
is continuous:
#
has non-empty topological interior (in
).
#
is
complete and
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
and
#
is
bornological and
is a
semi-complete strict -cone in
#
is the
inductive limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any catego ...
of a family
of ordered
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s with respect to a family of positive linear maps where
for all
where
is the positive cone of
Continuous positive extensions
The following theorem is due to H. Bauer and independently, to Namioka.
:Theorem: Let
be an
ordered topological vector space (TVS) with positive cone
let
be a vector subspace of
and let
be a linear form on
Then
has an extension to a continuous positive linear form on
if and only if there exists some convex neighborhood
of
in
such that
is bounded above on
:Corollary: Let
be an
ordered topological vector space with positive cone
let
be a vector subspace of
If
contains an interior point of
then every continuous positive linear form on
has an extension to a continuous positive linear form on
:Corollary: Let
be an
ordered vector space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition
Given a vector space ''X'' over the real numbers R and a pr ...
with positive cone
let
be a vector subspace of
and let
be a linear form on
Then
has an extension to a positive linear form on
if and only if there exists some convex
absorbing subset in
containing the origin of
such that
is bounded above on
Proof: It suffices to endow
with the finest locally convex topology making
into a neighborhood of
Examples
Consider, as an example of
the C*-algebra of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
with the positive elements being the
positive-definite matrices. The
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
function defined on this C*-algebra is a positive functional, as the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of any positive-definite matrix are positive, and so its trace is positive.
Consider the
Riesz space
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice.
Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Su ...
of all
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
complex-valued functions of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
support on a
locally compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
Consider a
Borel regular measure Borel may refer to:
People
* Borel (author), 18th-century French playwright
* Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance
* Émile Borel (1871 – 1956), a French mathematician known for his founding work in the are ...
on
and a functional
defined by
Then, this functional is positive (the integral of any positive function is a positive number). Moreover, any positive functional on this space has this form, as follows from the
Riesz–Markov–Kakutani representation theorem
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuo ...
.
Positive linear functionals (C*-algebras)
Let
be a C*-algebra (more generally, an
operator system in a C*-algebra
) with identity
Let
denote the set of positive elements in
A linear functional
on
is said to be if
for all
:Theorem. A linear functional
on
is positive if and only if
is bounded and
Cauchy–Schwarz inequality
If
is a positive linear functional on a C*-algebra
then one may define a semidefinite
sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
on
by
Thus from the
Cauchy–Schwarz inequality we have
Applications to economics
Given a space
, a price system can be viewed as a continuous, positive, linear functional on
.
See also
*
*
References
Bibliography
*
, ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. .
*
*
*
{{DEFAULTSORT:Positive Linear Functional
Functional analysis