In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the spaces of test functions and distributions are
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
s (TVSs) that are used in the definition and application of
distributions.
Test functions are usually
infinitely differentiable complex-valued (or sometimes
real-valued) functions on a non-empty
open subset that have
compact support.
The space of all test functions, denoted by
is endowed with a certain topology, called the , that makes
into a
complete Hausdorff locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
TVS.
The
strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded su ...
of
is called and is denoted by
where the "
" subscript indicates that the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of
denoted by
is endowed with the
strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
.
There are other possible choices for the space of test functions, which lead to other different
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
s of distributions. If
then the use of
Schwartz functions[The Schwartz space consists of smooth rapidly decreasing test functions, where "rapidly decreasing" means that the function decreases faster than any polynomial increases as points in its domain move away from the origin.] as test functions gives rise to a certain subspace of
whose elements are called . These are important because they allow the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a
vector subspace of the space of distributions
and is thus one example of a space of distributions; there are many other spaces of distributions.
There also exist other major classes of test functions that are subsets of
such as spaces of
analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support.
[Except for the trivial (i.e. identically ) map, which of course is always analytic.] Use of analytic test functions leads to
Sato's theory of
hyperfunctions.
Notation
The following notation will be used throughout this article:
*
is a fixed positive integer and
is a fixed non-empty
open subset of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
*
denotes the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s.
*
will denote a non-negative integer or
* If
is a
function then
will denote its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
and the of
denoted by
is defined to be the
closure of the set
in
* For two functions
, the following notation defines a canonical
pairing:
* A of size
is an element in
(given that
is fixed, if the size of multi-indices is omitted then the size should be assumed to be
). The of a multi-index
is defined as
and denoted by
Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index
:
We also introduce a partial order of all multi-indices by
if and only if
for all
When
we define their multi-index binomial coefficient as:
*
will denote a certain non-empty collection of compact subsets of
(described in detail below).
Definitions of test functions and distributions
In this section, we will formally define real-valued distributions on . With minor modifications, one can also define complex-valued distributions, and one can replace
with any (
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norm ...
)
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
.

Note that for all
and any compact subsets and of , we have:
Distributions on are defined to be the
continuous linear functionals on
when this vector space is endowed with a particular topology called the .
This topology is unfortunately not easy to define but it is nevertheless still possible to characterize distributions in a way so that no mention of the canonical LF-topology is made.
Proposition: If 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 , th ...
on
then the is a distribution if and only if the following equivalent conditions are satisfied:
# For every compact subset
there exist constants
and
(dependent on
) such that for all
# For every compact subset
there exist constants
and
such that for all
with
support contained in
[See for example .]
# For any compact subset
and any sequence
in
if
converges uniformly to zero on
for all
multi-indices , then
The above characterizations can be used to determine whether or not a linear functional is a distribution, but more advanced uses of distributions and test functions (such as applications to
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
) is limited if no topologies are placed on
and
To define the space of distributions we must first define the canonical LF-topology, which in turn requires that several other
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
s (TVSs) be defined first. First, a (
non-normable) topology on
will be defined, then every
will be endowed with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
induced on it by
and finally the (
non-metrizable) canonical LF-topology on
will be defined.
The space of distributions, being defined as the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of
is then endowed with the (non-metrizable)
strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
induced by
and the canonical LF-topology (this topology is a generalization of the usual
operator norm induced topology that is placed on the continuous dual spaces of
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "lengt ...
s).
This finally permits consideration of more advanced notions such as convergence of distributions (both sequences nets), various (sub)spaces of distributions, and operations on distributions, including extending differential equations to distributions.
Choice of compact sets K
Throughout,
will be any collection of compact subsets of
such that (1)
and (2) for any compact
there exists some
such that
The most common choices for
are:
* The set of all compact subsets of
or
* A set
where
and for all ,
and
is a
relatively compact non-empty open subset of
(here, "relatively compact" means that the
closure of
in either or
is compact).
We make
into a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
by defining
if and only if
Note that although the definitions of the subsequently defined topologies explicitly reference
in reality they do not depend on the choice of
that is, if
and
are any two such collections of compact subsets of
then the topologies defined on
and
by using
in place of
are the same as those defined by using
in place of
Topology on ''C''''k''(''U'')
We now introduce the
seminorms that will define the topology on
Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.
All of the functions above are non-negative
-valued
[The image of the compact set under a continuous -valued map (for example, for ) is itself a compact, and thus bounded, subset of If then this implies that each of the functions defined above is -valued (that is, none of the supremums above are ever equal to ).] seminorms on
As explained in
this article, every set of seminorms on a vector space induces a
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
vector topology.
Each of the following sets of seminorms
generate the same
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
vector topology on
(so for example, the topology generated by the seminorms in
is equal to the topology generated by those in
).
With this topology,
becomes a locally convex
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 ...
that is
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
. Every element of
is a continuous seminorm on
Under this topology, a
net in
converges to
if and only if for every multi-index
with
and every compact
the net of partial derivatives
converges uniformly
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitra ...
to
on
For any
any
(von Neumann) bounded subset of
is a
relatively compact subset of
In particular, a subset of
is bounded if and only if it is bounded in
for all
The space
is a
Montel space if and only if
The topology on
is the superior limit of the
subspace topologies induced on
by the TVSs
as ranges over the non-negative integers. A subset
of
is open in this topology if and only if there exists
such that
is open when
is endowed with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
induced on it by
Metric defining the topology
If the family of compact sets
satisfies
and
for all
then a complete translation-invariant metric on
can be obtained by taking a suitable countable
Fréchet combination
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
of any one of the above families. For example, using the seminorms
results in the metric
Often, it is easier to just consider seminorms.
Topology on ''C''''k''(''K'')
As before, fix
Recall that if
is any compact subset of
then
For any compact subset
is a closed subspace of the Fréchet space
and is thus also a
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 ...
. For all compact
satisfying
denote the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
by
Then this map is a linear embedding of TVSs (that is, it is a linear map that is also a
topological embedding) whose
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
(or "range") is closed in its
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
; said differently, the topology on
is identical to the subspace topology it inherits from
and also
is a closed subset of
The
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of
relative to
is empty.
If
is finite then
is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
with a topology that can be defined by the
norm
And when
then
is even a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
. The space
is a
distinguished Schwartz
Schwartz may refer to:
*Schwartz (surname), a surname (and list of people with the name)
*Schwartz (brand), a spice brand
*Schwartz's, a delicatessen in Montreal, Quebec, Canada
*Schwartz Publishing, an Australian publishing house
*"Danny Schwartz" ...
Montel space so if
then it is
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
and thus a Banach space (although like all other
it is a
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 ...
).
Trivial extensions and independence of ''C''''k''(''K'')'s topology from ''U''
The definition of
depends on so we will let
denote the topological space
which by definition is a
topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
of
Suppose
is an open subset of
containing
and for any compact subset
let
is the vector subspace of
consisting of maps with
support contained in
Given
its is by definition, the function
defined by:
so that
Let
denote the map that sends a function in
to its trivial extension on . This map is a linear
injection and for every compact subset
(where
is also a compact subset of
since
) we have
If is restricted to
then the following induced linear map is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
(and thus a
TVS-isomorphism
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is a ...
):
and thus the next two maps (which like the previous map are defined by
) are
topological embeddings:
(the topology on
is the canonical LF topology, which is defined later).
Using the injection
the vector space
is canonically identified with its image in
(however, if
then
is a
topological embedding when these spaces are endowed with their canonical LF topologies, although it is continuous).
Because
through this identification,
can also be considered as a subset of
Importantly, the subspace topology
inherits from
(when it is viewed as a subset of
) is identical to the subspace topology that it inherits from
(when
is viewed instead as a subset of
via the identification). Thus the topology on
is independent of the open subset of
that contains . This justifies the practice of written
instead of
Canonical LF topology
Recall that
denote all those functions in
that have compact
support in
where note that
is the union of all
as ranges over
Moreover, for every ,
is a dense subset of
The special case when
gives us the space of test functions.
This section defines the canonical LF topology as a
direct limit. It is also possible to define this topology in terms of its neighborhoods of the origin, which is described afterwards.
Topology defined by direct limits
For any two sets and , we declare that
if and only if
which in particular makes the collection
of compact subsets of into a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
(we say that such a collection is ). For all compact
satisfying
there are
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
s
Recall from above that the map
is a
topological embedding. The collection of maps
forms a
direct system
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors fr ...
in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topologica ...
s that is
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
by
(under subset inclusion). This system's
direct limit (in the category of locally convex TVSs) is the pair
where
are the natural inclusions and where
is now endowed with the (unique)
strongest
"Strongest" is a song recorded by Norwegian singer and songwriter Ina Wroldsen. The song was released on 27 October 2017 and has peaked at number 2 in Norway.
"Strongest" is Wroldsen's first solo release on Syco Music
Syco Music is a defunc ...
locally convex topology making all of the inclusion maps
continuous.
Topology defined by neighborhoods of the origin
If is a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
subset of
then is a
neighborhood of the origin in the canonical LF topology if and only if it satisfies the following condition:
Note that any convex set satisfying this condition is necessarily
absorbing in
Since the topology of any
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually the canonical LF topology by declaring that a convex
balanced subset is a neighborhood of the origin if and only if it satisfies condition .
Topology defined via differential operators
A is a sum
where
and all but finitely many of
are identically . The integer
is called the of the differential operator
If
is a linear differential operator of order then it induces a canonical linear map
defined by
where we shall reuse notation and also denote this map by
For any
the canonical LF topology on
is the weakest locally convex TVS topology making all linear differential operators in
of order
into continuous maps from
into
Properties of the canonical LF topology
=Canonical LF topology's independence from
=
One benefit of defining the canonical LF topology as the direct limit of a
direct system
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors fr ...
is that we may immediately use the universal property of direct limits. Another benefit is that we can use well-known results from
category theory to deduce that the canonical LF topology is actually independent of the particular choice of the
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
collection
of compact sets. And by considering different collections
(in particular, those
mentioned at the beginning of this article), we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes
into a
Hausdorff locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
strict LF-space (and also a
strict LB-space if
), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).
[If we take to be the set of compact subsets of then we can use the universal property of direct limits to conclude that the inclusion is a continuous and even that they are topological embedding for every compact subset If however, we take to be the set of closures of some countable increasing sequence of relatively compact open subsets of having all of the properties mentioned earlier in this in this article then we immediately deduce that is a Hausdorff locally convex strict LF-space (and even a strict LB-space when ). All of these facts can also be proved directly without using direct systems (although with more work).]
=Universal property
=
From the universal property of
direct limits, we know that if
is a linear map into a locally convex space (not necessarily Hausdorff), then is continuous if and only if is
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
if and only if for every
the restriction of to
is continuous (or bounded).
=Dependence of the canonical LF topology on
=
Suppose is an open subset of
containing
Let
denote the map that sends a function in
to its trivial extension on (which was defined above). This map is a continuous linear map. If (and only if)
then
is a dense subset of
and
is a
topological embedding. Consequently, if
then the transpose of
is neither one-to-one nor onto.
=Bounded subsets
=
A subset
is
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
in
if and only if there exists some
such that
and
is a bounded subset of
Moreover, if
is compact and
then
is bounded in
if and only if it is bounded in
For any
any bounded subset of
(resp.
) is a
relatively compact subset of
(resp.
), where
=Non-metrizability
=
For all compact
the interior of
in
is empty so that
is of the first category in itself. It follows from
Baire's theorem that
is
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 , \inf ...
and thus also
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
(see this footnote
[For any ]TVS TVS may refer to:
Mathematics
* Topological vector space
Television
* Television Sydney, TV channel in Sydney, Australia
* Television South, ITV franchise holder in the South of England between 1982 and 1992
* TVS Television Network, US dis ...
(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 , \inf ...
or otherwise), the notion of completeness depends entirely on a certain so-called "canonical uniformity
Uniformity may refer to:
* Distribution uniformity, a measure of how uniformly water is applied to the area being watered
* Religious uniformity, the promotion of one state religion, denomination, or philosophy to the exclusion of all other relig ...
" that is defined using the subtraction operation (see the article Complete topological vector space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x toward ...
for more details). In this way, the notion of a complete TVS
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
does not the existence of any metric. However, if the TVS is metrizable and if is translation-invariant metric on that defines its topology, then is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if is a complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bo ...
. So if a TVS happens to have a topology that can be defined by such a metric then may be used to deduce the completeness of but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ ...
of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on ). for an explanation of how the non-metrizable space
can be complete even though it does not admit a metric). The fact that
is a
nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
*Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
* Nuclear ...
Montel space makes up for the non-metrizability of
(see this footnote for a more detailed explanation).
[One reason for giving the canonical LF topology is because it is with this topology that and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of ]topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
s (TVSs) that are particularly similar to finite-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
s: the Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
s (especially Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s) and the nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
*Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
* Nuclear ...
Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
*Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
* Nuclear ...
Montel Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. differentiability, such as and ) end up being nuclear TVSs while TVSs associated with continuous differentiability (such as with compact and ) often end up being non-nuclear spaces, such as Banach spaces.
=Relationships between spaces
=
Using the
universal property of direct limits and the fact that the natural inclusions
are all
topological embedding, one may show that all of the maps
are also topological embeddings. Said differently, the topology on
is identical to the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
that it inherits from
where recall that
's topology was to be the subspace topology induced on it by
In particular, both
and
induces the same subspace topology on
However, this does imply that the canonical LF topology on
is equal to the subspace topology induced on
by
; these two topologies on
are in fact equal to each other since the canonical LF topology is metrizable while the subspace topology induced on it by
is metrizable (since recall that
is metrizable). The canonical LF topology on
is actually than the subspace topology that it inherits from
(thus the natural inclusion
is continuous but a
topological embedding).
Indeed, the canonical LF topology is so
fine
Fine may refer to:
Characters
* Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny''
* Officer Fine, a character in ''Tales from the Crypt'', played by Vincent Spano
Legal terms
* Fine (penalty), money to be paid as punishment for an offe ...
that if
denotes some linear map that is a "natural inclusion" (such as
or
or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions,
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on
the fine nature of the canonical LF topology means that more linear functionals on
end up being continuous ("more" means as compared to a coarser topology that we could have placed on
such as for instance, the subspace topology induced by some
which although it would have made
metrizable, it would have also resulted in fewer linear functionals on
being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making
into a
complete TVS
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
).
=Other properties
=
* The differentiation map
is a surjective continuous linear operator.
* The
bilinear multiplication map given by
is continuous; it is however,
hypocontinuous.
Distributions
As discussed earlier, continuous
linear functionals on a
are known as distributions on . Thus the set of all distributions on is the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of
which when endowed with the
strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
is denoted by
We have the canonical
duality pairing
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** ...
between a distribution on and a test function
which is denoted using
angle brackets
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
by
One interprets this notation as the distribution acting on the test function
to give a scalar, or symmetrically as the test function
acting on the distribution .
Characterizations of distributions
Proposition. If 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 , th ...
on
then the following are equivalent:
# is a distribution;
# : is a
continuous function.
# is
continuous at the origin.
# is
uniformly continuous.
# is 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 vecto ...
.
# is
sequentially continuous.
#* explicitly, for every sequence
in
that converges in
to some
[Even though the topology of is not metrizable, a linear functional on is continuous if and only if it is sequentially continuous.]
# is
sequentially continuous at the origin; in other words, maps null sequences
[ to null sequences.
#* explicitly, for every sequence in that converges in to the origin (such a sequence is called a ),
#* a is by definition a sequence that converges to the origin.
# maps null sequences to bounded subsets.
#* explicitly, for every sequence in that converges in to the origin, the sequence is bounded.
# maps Mackey convergent null sequences][ to bounded subsets;
#* explicitly, for every Mackey convergent null sequence in the sequence is bounded.
#* a sequence is said to be if there exists a divergent sequence of positive real number such that the sequence is bounded; every sequence that is Mackey convergent to necessarily converges to the origin (in the usual sense).
# The kernel of is a closed subspace of
# The graph of is closed.
# There exists a continuous seminorm on such that
# There exists a constant a collection of continuous seminorms, that defines the canonical LF topology of and a finite subset such that ][If is also a ]directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
under the usual function comparison then we can take the finite collection to consist of a single element.
# For every compact subset there exist constants and such that for all
# For every compact subset there exist constants and such that for all with support contained in [
# For any compact subset and any sequence in if converges uniformly to zero for all multi-indices then
# Any of the statements immediately above (that is, statements 14, 15, and 16) but with the additional requirement that compact set belongs to
]
Topology on the space of distributions
The topology of uniform convergence on bounded subsets is also called .[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 ...
, the strong dual topology is often the "standard" or "default" topology placed on the continuous dual space where if is a normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "lengt ...
then this strong dual topology is the same as the usual norm-induced topology on This topology is chosen because it is with this topology that becomes a nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
*Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
* Nuclear ...
Montel space and it is with this topology that the kernels theorem of Schwartz holds. No matter what dual topology is placed on [Technically, the topology must be coarser than the strong dual topology and also simultaneously be finer that the weak* topology.] a of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net). No matter which topology is chosen, will be a non-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 , \inf ...
, locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
. The space is separable[ and has the ]strong Pytkeev property
Strong may refer to:
Education
* The Strong, an educational institution in Rochester, New York, United States
* Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas
* Strong School, New Haven, Connecticut, United St ...
[Gabriyelyan, S.S. Kakol J., and·Leiderman, A]
"The strong Pitkeev property for topological groups and topological vector spaces"
/ref> but it is neither a k-space[ nor a ]sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
,[ which in particular implies that it is not ]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 , \inf ...
and also that its topology can be defined using only sequences.
Topological properties
Topological vector space categories
The canonical LF topology makes into a complete distinguished strict LF-space (and a strict LB-space if and only if ), which implies that is a meager subset of itself. Furthermore, as well as its strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded su ...
, is a complete Hausdorff locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
barrelled bornological Mackey space. The strong dual of is a 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 ...
if and only if so in particular, the strong dual of which is the space of distributions on , is metrizable (note that the weak-* topology on also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
gives ).
The three spaces and the Schwartz space as well as the strong duals of each of these three spaces, are complete nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
*Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
* Nuclear ...
Montel bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
s, which implies that all six of these locally convex spaces are also paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norm ...
reflexive barrelled Mackey spaces. The spaces and are both distinguished 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 ...
s. Moreover, both and are Schwartz TVSs.
Convergent sequences
=Convergent sequences and their insufficiency to describe topologies
=
The strong dual spaces of and are sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
s but not Fréchet-Urysohn spaces.[ Moreover, neither the space of test functions nor its strong dual is a ]sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
(not even an Ascoli space Ascoli may refer to:
Places in Italy
*Ascoli Satriano, a town and ''comune'' in the province of Foggia in the Apulia region
*Province of Ascoli Piceno, a province of the Marche region
** Ascoli Piceno, a city which is the seat of the province above ...
),[Gabriyelyan, Saa]
"Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"
(2017)[T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.] which in particular implies that their topologies can be defined entirely in terms of convergent sequences.
A sequence in converges in if and only if there exists some such that contains this sequence and this sequence converges in ; equivalently, it converges if and only if the following two conditions hold:
# There is a compact set containing the supports of all
# For each multi-index the sequence of partial derivatives tends uniformly
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
to
Neither the space nor its strong dual is a sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
,[ and consequently, their topologies can be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is enough to define the canonical LF topology on The same can be said of the strong dual topology on
]
=What sequences do characterize
=
Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of any Montel space, a sequence converges in the strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
if and only if it converges in the weak* topology, which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually the convergence of a sequence of distributions; this is fine for sequences but it does extend to the convergence of nets of distributions since a net may converge pointwise but fail to converge in the strong dual topology).
Sequences characterize continuity of linear maps valued in locally convex space. Suppose is a locally convex bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
(such as any of the six TVSs mentioned earlier). Then a linear map into a locally convex space is continuous if and only if it maps null sequences[A is a sequence that converges to the origin.] in to bounded subsets of .[Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences.] More generally, such a linear map is continuous if and only if it maps Mackey convergent null sequences[A sequence is said to be if there exists a divergent sequence of positive real number such that is a bounded set in ] to bounded subsets of So in particular, if a linear map into a locally convex space is sequentially continuous at the origin then it is continuous. However, this does necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs.
For every is sequentially dense
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
in Furthermore, is a sequentially dense subset of (with its strong dual topology) and also a sequentially dense subset of the strong dual space of
=Sequences of distributions
=
A sequence of distributions converges with respect to the weak-* topology on to a distribution if and only if
for every test function For example, if is the function
and is the distribution corresponding to then
as so in Thus, for large the function can be regarded as an approximation of the Dirac delta distribution.
=Other properties
=
* The strong dual space of is TVS isomorphic to via the canonical TVS-isomorphism defined by sending to (that is, to the linear functional on defined by sending to );
* On any bounded subset of the weak and strong subspace topologies coincide; the same is true for ;
* Every weakly convergent sequence in is strongly convergent (although this does not extend to nets).
Localization of distributions
Preliminaries: Transpose of a linear operator
Operations on distributions and spaces of distributions are often defined by means of the transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well known 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 ...
.[; .] For instance, the well-known Hermitian adjoint of a linear operator between Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
). In general the transpose of a continuous linear map is the linear map
or equivalently, it is the unique map satisfying for all and all (the prime symbol in does not denote a derivative of any kind; it merely indicates that is an element of the continuous dual space ). Since is continuous, the transpose is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).
In the context of distributions, the characterization of the transpose can be refined slightly. Let be a continuous linear map. Then by definition, the transpose of is the unique linear operator that satisfies:
Since is dense in (here, actually refers to the set of distributions ) it is sufficient that the defining equality hold for all distributions of the form where Explicitly, this means that a continuous linear map is equal to if and only if the condition below holds:
where the right hand side equals
Extensions and restrictions to an open subset
Let be open subsets of
Every function can be from its domain to a function on by setting it equal to on the complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
This extension is a smooth compactly supported function called the and it will be denoted by
This assignment defines the operator
which is a continuous injective linear map. It is used to canonically identify as a vector subspace of (although as a topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
).
Its transpose ( explained here)
is called the and as the name suggests, the image of a distribution under this map is a distribution on called the restriction of to The defining condition of the restriction is:
If then the (continuous injective linear) trivial extension map is a topological embedding (in other words, if this linear injection was used to identify as a subset of then 's topology would strictly finer than the subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
that induces on it; importantly, it would be a topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
since that requires equality of topologies) and its range is also dense in its codomain Consequently, if then the restriction mapping is neither injective nor surjective. A distribution is said to be if it belongs to the range of the transpose of and it is called if it is extendable to
Unless the restriction to is neither injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
nor surjective.
Spaces of distributions
For all and all all of the following canonical injections are continuous and have an image/range that is a dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
of their codomain:
where the topologies on the LB-spaces are the canonical LF topologies as defined below (so in particular, they are not the usual norm topologies).
The range of each of the maps above (and of any composition of the maps above) is dense in the codomain. Indeed, is even sequentially dense
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
in every For every the canonical inclusion into the normed space (here has its usual norm topology) is a continuous linear injection and the range of this injection is dense in its codomain if and only if .
Suppose that is one of the LF-spaces (for ) or LB-spaces (for ) or normed spaces (for ). Because the canonical injection is a continuous injection whose image is dense in the codomain, this map's transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is a continuous injection. This injective transpose map thus allows the continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of to be identified with a certain vector subspace of the space of all distributions (specifically, it is identified with the image of this transpose map). This continuous transpose map is not necessarily a TVS-embedding so the topology that this map transfers from its domain to the image is finer than the subspace topology that this space inherits from
A linear subspace of carrying a locally convex topology that is finer than the subspace topology induced by is called .
Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order some integer, distributions induced by a positive Radon measure, distributions induced by an -function, etc.) and any representation theorem about the dual space of may, through the transpose be transferred directly to elements of the space
Compactly supported ''Lp''-spaces
Given the vector space of on and its topology are defined as direct limits of the spaces in a manner analogous to how the canonical LF-topologies on were defined.
For any compact let denote the set of all element in (which recall are equivalence class of Lebesgue measurable functions on ) having a representative whose support (which recall is the closure of in ) is a subset of (such an is almost everywhere defined in ).
The set is a closed vector subspace and is thus a Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
and when even a Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
.
Let be the union of all as ranges over all compact subsets of
The set is a vector subspace of whose elements are the (equivalence classes of) compactly supported functions defined on (or almost everywhere on ).
Endow with the final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that ...
(direct limit topology) induced by the inclusion maps as ranges over all compact subsets of
This topology is called the and it is equal to the final topology induced by any countable set of inclusion maps () where are any compact sets with union equal to
This topology makes into an LB-space (and thus also an LF-space) with a topology that is strictly finer than the norm (subspace) topology that induces on it.
Radon measures
The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is also a continuous injection.
Note that the continuous dual space can be identified as the space of Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s, where there is a one-to-one correspondence between the continuous linear functionals and integral with respect to a Radon measure; that is,
* if then there exists a Radon measure on such that for all and
* if is a Radon measure on then the linear functional on defined by is continuous.
Through the injection every Radon measure becomes a distribution on . If is a locally integrable function on then the distribution is a Radon measure; so Radon measures form a large and important space of distributions.
The following is the theorem of the structure of distributions of Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s, which shows that every Radon measure can be written as a sum of derivatives of locally functions in :
Positive Radon measures
A linear function on a space of functions is called if whenever a function that belongs to the domain of is non-negative (meaning that is real-valued and ) then One may show that every positive linear functional on is necessarily continuous (that is, necessarily a Radon measure).
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
is an example of a positive Radon measure.
Locally integrable functions as distributions
One particularly important class of Radon measures are those that are induced locally integrable functions. The function is called if it is Lebesgue integrable over every compact subset of .[For more information on such class of functions, see the entry on locally integrable functions.] This is a large class of functions which includes all continuous functions and all Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
functions. The topology on is defined in such a fashion that any locally integrable function yields a continuous linear functional on – that is, an element of – denoted here by , whose value on the test function is given by the Lebesgue integral:
Conventionally, one abuses notation by identifying with provided no confusion can arise, and thus the pairing between and is often written
If and are two locally integrable functions, then the associated distributions and are equal to the same element of if and only if and are equal almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
(see, for instance, ). In a similar manner, every Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
on defines an element of whose value on the test function is As above, it is conventional to abuse notation and write the pairing between a Radon measure and a test function as Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.
Test functions as distributions
The test functions are themselves locally integrable, and so define distributions. The space of test functions is sequentially dense in with respect to the strong topology on This means that for any there is a sequence of test functions, that converges to (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
Furthermore, is also sequentially dense in the strong dual space of
Distributions with compact support
The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is also a continuous injection. Thus the image of the transpose, denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so the topology of is finer than the subspace topology that this set inherits from ).
The elements of can be identified as the space of distributions with compact support. Explicitly, if is a distribution on then the following are equivalent,
* ;
* the support of is compact;
* the restriction of to when that space is equipped with the subspace topology inherited from (a coarser topology than the canonical LF topology), is continuous;
* there is a compact subset of such that for every test function whose support is completely outside of , we have
Compactly supported distributions define continuous linear functionals on the space ; recall that the topology on is defined such that a sequence of test functions converges to 0 if and only if all derivatives of converge uniformly to 0 on every compact subset of . Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from to
Distributions of finite order
Let The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is also a continuous injection. Consequently, the image of denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so 's topology is finer than the subspace topology that this set inherits from ). The elements of are The distributions of order which are also called are exactly the distributions that are Radon measures (described above).
For a is a distribution of order that is not a distribution of order
A distribution is said to be of if there is some integer such that it is a distribution of order and the set of distributions of finite order is denoted by Note that if then so that is a vector subspace of and furthermore, if and only if
Structure of distributions of finite order
Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if is the restriction mapping from to , then the image of under is contained in
The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s:
Example. (Distributions of infinite order) Let and for every test function let
Then is a distribution of infinite order on . Moreover, can not be extended to a distribution on ; that is, there exists no distribution on such that the restriction of to is equal to .
Tempered distributions and Fourier transform
Defined below are the , which form a subspace of the space of distributions on This is a proper subspace: while every tempered distribution is a distribution and an element of the converse is not true. Tempered distributions are useful if one studies the Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in
Schwartz space
The Schwartz space, is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus is in the Schwartz space provided that any derivative of multiplied with any power of converges to 0 as These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices and define:
Then is in the Schwartz space if all the values satisfy:
The family of seminorms defines a locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
topology on the Schwartz space. For the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:
Otherwise, one can define a norm on via
The Schwartz space is a 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 ...
(i.e. a complete 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 , \inf ...
locally convex space). Because the Fourier transform changes into multiplication by and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence in converges to 0 in if and only if the functions converge to 0 uniformly in the whole of which implies that such a sequence must converge to zero in
is dense in The subset of all analytic Schwartz functions is dense in as well.
The Schwartz space is nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
*Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
* Nuclear ...
and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
where represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product).
Tempered distributions
The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is also a continuous injection. Thus, the image of the transpose map, denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so the topology of is finer than the subspace topology that this set inherits from ).
The space is called the space of . It is the continuous dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
of the Schwartz space. Equivalently, a distribution is a tempered distribution if and only if
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
for are tempered distributions.
The can also be characterized as , meaning that each derivative of grows at most as fast as some polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of decays faster than every inverse power of An example of a rapidly falling function is for any positive
Fourier transform
To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform is a TVS- automorphism of the Schwartz space, and the is defined to be its transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
which (abusing notation) will again be denoted by . So the Fourier transform of the tempered distribution is defined by for every Schwartz function is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
and also with convolution: if is a tempered distribution and is a smooth function on is again a tempered distribution and
is the convolution of and . In particular, the Fourier transform of the constant function equal to 1 is the distribution.
Expressing tempered distributions as sums of derivatives
If is a tempered distribution, then there exists a constant and positive integers and such that for all Schwartz functions
This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function and a multi-index such that
Restriction of distributions to compact sets
If then for any compact set there exists a continuous function compactly supported in (possibly on a larger set than itself) and a multi-index such that on
Tensor product of distributions
Let and be open sets. Assume all vector spaces to be over the field where or For define for every and every the following functions:
Given and define the following functions:
where and
These definitions associate every and with the (respective) continuous linear map:
Moreover, if either (resp. ) has compact support then it also induces a continuous linear map of (resp.
denoted by or is the distribution in defined by:
Schwartz kernel theorem
The tensor product defines a bilinear map
the span of the range of this map is a dense subspace of its codomain. Furthermore, Moreover induces continuous bilinear maps:
where denotes the space of distributions with compact support and is the Schwartz space of rapidly decreasing functions.
This result does not hold for Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s such as and its dual space. Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space ? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because is a nuclear space that the Schwartz kernel theorem holds. Like Hilbert spaces, nuclear spaces may be thought as of generalizations of finite dimensional Euclidean space.
Using holomorphic functions as test functions
The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
and several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.
See also
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Notes
References
Bibliography
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Further reading
* M. J. Lighthill (1959). ''Introduction to Fourier Analysis and Generalised Functions''. Cambridge University Press. (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
* V.S. Vladimirov (2002). ''Methods of the theory of generalized functions''. Taylor & Francis.
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{{Topological vector spaces
Functional analysis
Generalized functions
Generalizations of the derivative
Smooth functions
Topological vector spaces