Mathematical analysis term similar to generalized function
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.
A function
is normally thought of as acting on the points in its domain by "sending" a point x in its domain to the point
Instead of acting on points, distribution theory reinterprets functions such as
as acting on test functions in a certain way. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions with compact support (bump functions are examples of test functions). Many "standard functions" (meaning for example a function that is typically encountered in a Calculus course), say for instance a continuous map
can be canonically reinterpreted as acting on test functions (instead of their usual interpretation as acting on points of their domain) via the action known as "integration against a test function"; explicitly, this means that
"acts on" a test function g by "sending" g to the number
This new action of
is thus a complex (or real)-valued map, denoted by
whose domain is the space of test functions; this map turns out to have two additional properties[note 1] that make it into what is known as a distribution on
Distributions that arise from "standard functions" in this way are the prototypical examples of a distributions. But there are many distributions that do not arise in this way and these distributions are known as "generalized functions." Examples include the Dirac delta function or some distributions that arise via the action of "integration of test functions against measures." However, by using various methods it is nevertheless still possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.
In applications to physics and engineering, the space of test functions usually consists of smooth functions with compact support that are defined on some given non-empty open subset
This space of test functions is denoted by
or
and a distribution on U is by definition a linear functional on
that is continuous when
is given a topology called the canonical LF topology. This leads to the space of (all) distributions on U, usually denoted by
(note the prime), which by definition is the space of all distributions on
(that is, it is the continuous dual space of
); it is these distributions that are the main focus of this article.
There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If
then the use of Schwartz functions[note 2] as test functions gives rise to a certain subspace of
whose elements are called tempered distributions. These are important because they allow the Fourier transform 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 not 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.[note 3] Use of analytic test functions lead to Sato's theory of hyperfunctions.
History
The practical use of distributions can be traced back to the use of Green functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.
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 
denotes the natural numbers.
will denote a non-negative integer or 
- If
is a function then
will denote its domain and the support of
denoted by
is defined to be the closure of the set
in 
- For two functions
, the following notation defines a canonical pairing:

- A multi-index 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 length 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 U. With minor modifications, one can also define complex-valued distributions, and one can replace
with any (paracompact) smooth manifold.
The graph of the
bump function 
where

and

This function is a test function on

and is an element of

The support of this function is the closed
unit disk in

It is non-zero on the open unit disk and it is equal to
0 everywhere outside of it.
Note that for all
and any compact subsets K and L of U, we have:

Distributions on U are defined to be the continuous linear functionals on
when this vector space is endowed with a particular topology called the canonical LF-topology.
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 T is a linear functional on
then the T is a distribution if and only if the following are equivalent conditions are satisfied:
- 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
[2]
- 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) 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 topological vector spaces (TVSs) be defined first. We will first define a topology on
then assign every
the subspace topology induced on it by
and finally we define the canonical LF-topology on
We use the canonical LF-topology to define a topology on the space of distributions, which permits us to consider things such as convergence of distributions.
- Choice of compact sets K
Throughout, K will be any collection of compact subsets of U such that (1)
and (2) for any compact K ? U there exists some K2 ? K such that K ? K2. The most common choices for K are:
- The set of all compact subsets of U, or
- A set
where
and for all i,
and Ui is a relatively compact non-empty open subset of U (i.e. "relatively compact" means that the closure of Ui, in either U or
is compact).
We make K into a directed set by defining K1K2 if and only if K1 ? K2. Note that although the definitions of the subsequently defined topologies explicitly reference K, in reality they do not depend on the choice of K; that is, if K1 and K2 are any two such collections of compact subsets of U, then the topologies defined on
and
by using K1 in place of K are the same as those defined by using K2 in place of K.
Topology on Ck(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.
Each of the functions above are non-negative R-valued[note 5]seminorms on
Each of the following families of seminorms generates the same locally convex vector topology on
:

With this topology,
becomes a locally convex (non-normable) Fréchet space and all of the seminorms defined above are continuous on this space. All of the seminorms defined above are continuous functions on
Under this topology, a net
in
converges to
if and only if for every multi-index p with || < k + 1 and every K ? K, the net
converges to
uniformly on K. For any
any 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 k = ?.
The topology on
is the superior limit of the subspace topologies induced on
by the TVSs
as i ranges over the non-negative integers. A subset W of
is open in this topology if and only if there exists
such that W is open when
is endowed with the subspace topology induced by
- Metric defining the topology
If the family of compact sets
satisfies
and
for all i, then a complete translation-invariant metric on
can be obtained by taking a suitable countable Fréchet combination of any one of the above families.
For example, using the seminorms
results in
![{\displaystyle d(f,g):=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {r_{i,{\overline {U_{i}}}}(f-g)}{1+r_{i,{\overline {U_{i}}}}(f-g)}}=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {\sup _{|p|\leq i,x\in {\overline {U_{i}}}}\left|\partial ^{p}(f-g)(x)\right|}{\left[1+\sup _{|p|\leq i,x\in {\overline {U_{i}}}}\left|\partial ^{p}(f-g)(x)\right|\right]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f612a1a2a41b8d23cf7204b4ab4ef26fb00cbf)
Often, it is easier to just consider seminorms.
Topology on Ck(K)
As before, fix
Recall that if
is any compact subset of
then
For any compact subset K ? U,
is a closed subspace of the Fréchet space
and is thus also a Fréchet space. For all compact K, L ? U with K ? L, denote the natural inclusion by
Then this map is a linear embedding of TVSs (i.e. a linear map that is also a topological embedding) whose range is closed in its codomain; said differently, the topology on
is identical to the subspace topology it inherits from
and also
is a closed subset of
The interior of
relative to
is empty.
If
is finite then
is a Banach space with a topology that can be defined by the norm

And when k = 2, then
is even a Hilbert space. The space
is a distinguished Schwartz Montel space so if
then it is not normable and thus not a Banach space (although like all other
it is a Fréchet space).
Trivial extensions and independence of Ck(K)'s topology from U
The definition of
depends on U so we will let
denote the topological space
which by definition is a topological subspace of
Suppose V is an open subset of
containing
Given
its trivial extension to V is by definition, the function
defined by:

so that
Let
denote the map that sends a function in
to its trivial extension on V. This map is a linear injection and for every compact subset
we have
where
is the vector subspace of
consisting of maps with support contained in K (since K ? U ? V, K is a compact subset of V as well). It follows that
If I is restricted to
then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism):

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
we identify
with its image in
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 U of
that contains K. This justifies the practice of written
instead of
Topology on the spaces of test functions and distributions
Recall that
denote all those functions in
that have compact support in U, where note that
is the union of all
as K ranges over K. Moreover, for every k,
is a dense subset of
The special case when k = ? gives us the space of test functions.
Canonical LF topology
We now define 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.
For any two sets K and L, we declare that K L if and only if K ? L, which in particular makes the collection K of compact subsets of U into a directed set (we say that such a collection is directed by subset inclusion). For all compact K, L ? U with K ? L, there are natural inclusions

Recall from above that the map
is a topological embedding. The collection of maps

forms a direct system in the category of locally convex topological vector spaces that is directed by K (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 locally convex topology making all of the inclusion maps
continuous.
- Neighborhoods of the origin
If U is a convex subset of
then U 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 is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually define the canonical LF topology by declaring that a convex balanced subset U is a neighborhood of the origin if and only if it satisfies condition CN.
- Topology defined via differential operators
A linear differential operator in U with smooth coefficients is a sum

where
and all but finitely many of
are identically 0. The integer
is called the order of the differential operator
If
is a linear differential operator of order k then it induces a canonical linear map
defined by
where we shall reuse notation and also denote this map by 
For any 1 k , the canonical LF topology on
is the weakest locally convex TVS topology making all linear differential operators in U of order < k + 1 into continuous maps from
into 
Basic properties
- Canonical LF topology's independence from K
One benefit of defining the canonical LF topology as the direct limit of a direct system 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 collection K of compact sets. And by considering different collections K (in particular, those K 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 strict LF-space (and also a strict LB-space if k ? ?), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).[note 6]
- Universal property
From the universal property of direct limits, we know that if
is a linear map into a locally convex space Y (not necessarily Hausdorff), then u is continuous if and only if u is bounded if and only if for every K ? K, the restriction of u to
is continuous (or bounded).
- Dependence of the canonical LF topology on U
Suppose V is an open subset of
containing
Let
denote the map that sends a function in
to its trivial extension on V (which was defined above). This map is a continuous linear map. If (and only if) U ? V then
is not a dense subset of
and
is not a topological embedding. Consequently, if U ? V then the transpose of
is neither one-to-one nor onto.
- Bounded subsets
A subset B of
is bounded in
if and only if there exists some K ? K such that
and B is a bounded subset of
Moreover, if K ? U is compact and
then S is bounded in
if and only if it is bounded in
For any 0 k , any bounded subset of
(resp.
) is a relatively compact subset of
(resp.
), where ? + 1 = ?.
- Non-metrizability
For all compact K ? U, the interior of
in
is empty so that
is of the first category in itself. It follows from Baire's theorem that
is not metrizable and thus also not normable (see this footnote[note 7] for an explanation of how the non-metrizable space
can be complete even thought it does not admit a metric). The fact that
is a nuclear Montel space makes up for the non-metrizability of
(see this footnote for a more detailed explanation).[note 8]
- 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 that it inherits from
where recall that
's topology was defined to be the subspace topology induced on it by
In particular, both
and
induces the same subspace topology on
However, this does not imply that the canonical LF topology on
is equal to the subspace topology induced on
by
; these two topologies on
are in fact never equal to each other since the canonical LF topology is never metrizable while the subspace topology induced on it by
is metrizable (since recall that
is metrizable). The canonical LF topology on
is actually strictly finer than the subspace topology that it inherits from
(thus the natural inclusion
is continuous but not a topological embedding).
Indeed, the canonical LF topology is so fine 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 measures, 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).
- Other properties
- The differentiation map
is a surjective continuous linear operator.
- The bilinear multiplication map
given by
is not continuous; it is however, hypocontinuous.
Distributions
As discussed earlier, continuous linear functionals on a
are known as distributions on U. Thus the set of all distributions on U is the continuous dual space of
which when endowed with the strong dual topology is denoted by
We have the canonical duality pairing between a distribution T on U and a test function
which is denoted using angle brackets by

One interprets this notation as the distribution T acting on the test function
to give a scalar, or symmetrically as the test function
acting on the distribution T.
- Characterizations of distributions
Proposition. If T is a linear functional on
then the following are equivalent:
- T is a distribution;
- (definition) T is continuous;
- T is continuous at the origin;
- T is uniformly continuous;
- T is a bounded operator;
- T is sequentially continuous;
- explicitly, for every sequence
in
that converges in
to some
[note 9]
- T is sequentially continuous at the origin; in other words, T maps null sequences[note 10] to null sequences;
- explicitly, for every sequence
in
that converges in
to the origin (such a sequence is called a null sequence), 
- a null sequence is by definition a sequence that converges to the origin;
- T maps null sequences to bounded subsets;
- explicitly, for every sequence
in
that converges in
to the origin, the sequence
is bounded;
- T maps Mackey convergence null sequences[note 11] to bounded subsets;
- explicitly, for every Mackey convergent null sequence
in
the sequence
is bounded;
- a sequence fo = (fi)?
i=1 is said to be Mackey convergent to 0 if there exists a divergent sequence ro = (ri)?
i=1 -> ? of positive real number such that the sequence (rifi)?
i=1 is bounded; every sequence that is Mackey convergent to 0 necessarily converges to the origin (in the usual sense);
- The kernel of T is a closed subspace of

- The graph of T is a closed;
- There exists a continuous seminorm g on
such that 
- There exists a constant C > 0, a collection of continuous seminorms,
that defines the canonical LF topology of
and a finite subset
such that
[note 12]
- 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
[15]
- For any compact subset
and any sequence
in
if
converges uniformly to zero for all multi-indices p, then 
- Any of the three statements immediately above (i.e. statements 14, 15, and 16) but with the additional requirement that compact set K belongs to K.
Topology on the space of distributions
The topology of uniform convergence on bounded subsets is also called the strong dual topology.[note 13] This topology is chosen because it is with this topology that
becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.[16] No matter what dual topology is placed on
,[note 14] a sequence 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, locally convex topological vector space. The space
is separable[17] and has the strong Pytkeev property[18] but it is neither a k-space[18] nor a sequential space,[17] which in particular implies that it is not metrizable and also that its topology can not 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 k ? ?), which implies that
is a meager subset of itself. Furthermore,
as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of
is a Fréchet space if and only if k ? ? so in particular, the strong dual of
which is the space
of distributions on U, is not 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 gives
).
The three spaces
and the Schwartz space
as well as the strong duals of each of these three spaces, are complete nuclearMontelbornological spaces, which implies that all six of these locally convex spaces are also paracompact[23]reflexive barrelled Mackey spaces. The spaces
and
are both distinguished Fréchet spaces. Moreover, both
and
are Schwartz TVSs.
Convergent sequences
- Convergent sequences and their insufficiency to describe topologies
The strong dual spaces of
and
are sequential spaces but not Fréchet-Urysohn spaces.[17] Moreover, neither the space of test functions
nor its strong dual
is a sequential space (not even an Ascoli space),[17][24] which in particular implies that their topologies can not be defined entirely in terms of convergent sequences.
A sequence
in
converges in
if and only if there exists some K ? K such that
contains this sequence and this sequence converges in
; equivalently, it converges if and only if the following two conditions hold:[25]
- There is a compact set K ? U containing the supports of all

- For each multi-index ?, the sequence of partial derivatives
tends uniformly to 
Neither the space
nor its strong dual
is a sequential space,[17][24] and consequently, their topologies can not be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is not 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 of any Montel space, a sequence converges in the strong dual topology 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 define the convergence of a sequence of distributions; this is fine for sequences but it does not extend to the convergence of nets of distributions since a net may converge pointwise but fail to convergence in the strong dual topology).
Sequences characterize continuity of linear maps valued in locally convex space. Suppose X is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map F : X -> Y into a locally convex space Y is continuous if and only if it maps null sequences[note 10] in X to bounded subsets of Y.[note 15] More generally, such a linear map F : X -> Y is continuous if and only if it maps Mackey convergent null sequences[note 11] to bounded subsets of
So in particular, if a linear map F : X -> Y into a locally convex space is sequentially continuous at the origin then it is continuous. However, this does not 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
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 T if and only if

for every test function
For example, if
is the function
![{\displaystyle f_{m}(x)={\begin{cases}m&{\text{if }}x\in [0,{\frac {1}{m}}]\\0&{\text{otherwise}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaa43d3ea88c00d8ef38c89536f943bbf3400da)
and Tm is the distribution corresponding to
then

as m -> ?, so Tm -> ? in
Thus, for large m, 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 value at
(i.e. 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
There is no way to define the value of a distribution in
at a particular point of U. However, as is the case with functions, distributions on U restrict to give distributions on open subsets of U. Furthermore, distributions are locally determined in the sense that a distribution on all of U can be assembled from a distribution on an open cover of U satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.
Restrictions to an open subset
Let U and V be open subsets of
with V ? U. Let
be the operator which extends by zero a given smooth function compactly supported in V to a smooth function compactly supported in the larger set U. The transpose of
is called the restriction mapping and is denoted by
The map
is a continuous injection where if V ? U then it is not a topological embedding and its range is not dense in
which implies that this map's transpose is neither injective nor surjective and that the topology that
transfers from
onto its image is strictly finer than the subspace topology that
induces on this same set. A distribution
is said to be extendible to U if it belongs to the range of the transpose of
and it is called extendible if it is extendable to 
For any distribution
the restriction ?VU(T) is a distribution in
defined by:

Unless U = V, the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if U = R and V = (0, 2), then the distribution

is in
but admits no extension to
Gluing and distributions that vanish in a set
Let V be an open subset of U.
is said to vanish in V if for all
such that
we have
T vanishes in V if and only if the restriction of T to V is equal to 0, or equivalently, if and only if T lies in the kernel of the restriction map ?VU.
- Corollary. Let
be a collection of open subsets of
and let
T = 0 if and only if for each
the restriction of T to
is equal to 0.
- Corollary. The union of all open subsets of U in which a distribution T vanishes is an open subset of U in which T vanishes.
Support of a distribution
This last corollary implies that for every distribution T on U, there exists a unique largest subset V of U such that T vanishes in V (and does not vanish in any open subset of U that is not contained in V); the complement in U of this unique largest open subset is called the support of T. Thus

If
is a locally integrable function on U and if
is its associated distribution, then the support of
is the smallest closed subset of U in the complement of which
is almost everywhere equal to 0. If
is continuous, then the support of
is equal to the closure of the set of points in U at which
does not vanish. The support of the distribution associated with the Dirac measure at a point
is the set
If the support of a test function
does not intersect the support of a distribution T then Tf = 0. A distribution T is 0 if and only if its support is empty. If
is identically 1 on some open set containing the support of a distribution T then fT = T. If the support of a distribution T is compact then it has finite order and furthermore, there is a constant C and a non-negative integer N such that:

If T has compact support then it has a unique extension to a continuous linear functional
on
; this functional can be defined by
where
is any function that is identically 1 on an open set containing the support of T.
If
and
then
and
Thus, distributions with support in a given subset
form a vector subspace of
; such a subspace is weakly closed in
if and only if A is closed in U. Furthermore, if
is a differential operator in U, then for all distributions T on U and all
we have
and 
Distributions with compact support
- Support in a point set and Dirac measures
For any
let
denote the distribution induced by the Dirac measure at x. For any
and distribution
the support of T is contained in
if and only if T is a finite linear combination of derivatives of the Dirac measure at
If in addition the order of T is
then there exist constants
such that:

Said differently, if T has support at a single point
then T is in fact a finite linear combination of distributional derivatives of the ? function at P. That is, there exists an integer m and complex constants a? such that

where
is the translation operator.
- Distribution with compact support
- Distributions of finite order with support in an open subset
Theorem — Suppose T is a distribution on U with compact support K and let V be an open subset of U containing K. Since every distribution with compact support has finite order, take N to be the order of T and define
There exists a family of continuous functions
defined on U with support in V such that

where the derivatives are understood in the sense of distributions. That is, for all test functions
on U,

Global structure of distributions
The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of
(or the Schwartz space
for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.
- Distributions as sheafs
Decomposition of distributions as sums of derivatives of continuous functions
By combining the above results, one may express any distribution on U as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on U. In other words for arbitrary
we can write:

where
are finite sets of multi-indices and the functions
are continuous.
Theorem — Let T be a distribution on U. For every multi-index p there exists a continuous function gp on U such that
- any compact subset K of U intersects the support of only finitely many gp, and

Moreover, if T has finite order, then one can choose gp in such a way that only finitely many of them are non-zero.
Note that the infinite sum above is well-defined as a distribution. The value of T for a given
can be computed using the finitely many g? that intersect the support of
Operations on distributions
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if
is a linear map which is continuous with respect to the weak topology, then it is possible to extend A to a map
by passing to the limit.[note 16][][clarification needed]
Preliminaries: Transpose of a linear operator
Operations on distributions and spaces of distributions are often defined by means of the transpose of a linear operator because it provides a unified approach that the many definitions in the theory of distributions and because of its many well-known topological properties.[36] In general the transpose of a continuous linear map
is the linear map
defined by
or equivalently, it is the unique map satisfying
for all
and all
Since A 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 A is the unique linear operator
that satisfies:
for all
and all 
However, since the image of
is dense in
it is sufficient that the above equality hold for all distributions of the form
where
Explicitly, this means that the above condition holds if and only if the condition below holds:
for all 
Differential operators
Differentiation of distributions
Let
is the partial derivative operator
In order to extend
we compute its transpose:
![{\displaystyle {\begin{aligned}\langle {}^{t}A(D_{\psi }),\phi \rangle &=\int _{U}\psi (A\phi )\,dx&&{\text{(See above.)}}\\&=\int _{U}\psi {\frac {\partial \phi }{\partial x_{k}}}\,dx\\[4pt]&=-\int _{U}\phi {\frac {\partial \psi }{\partial x_{k}}}\,dx&&{\text{(integration by parts)}}\\[4pt]&=-\left\langle {\frac {\partial \psi }{\partial x_{k}}},\phi \right\rangle \\[4pt]&=-\langle A\psi ,\phi \rangle \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b27be7f96dbc3c227d42c27753c5012a2d1892c)
Therefore
Therefore the partial derivative of
with respect to the coordinate
is defined by the formula

With this definition, every distribution is infinitely differentiable, and the derivative in the direction
is a linear operator on
More generally, if
is an arbitrary multi-index, then the partial derivative
of the distribution
is defined by

Differentiation of distributions is a continuous operator on
this is an important and desirable property that is not shared by most other notions of differentiation.
If T is a distribution in R then

where
is the derivative of T and ?x is translation by x; thus the derivative of T may be viewed as a limit of quotients.
Differential operators acting on smooth functions
A linear differential operator in U with smooth coefficients acts on the space of smooth functions on
Given
we would like to define a continuous linear map,
that extends the action of
on
to distributions on
In other words we would like to define
such that the following diagram commutes:

Where the vertical maps are given by assigning
its canonical distribution
which is defined by:
for all
With this notation the diagram commuting is equivalent to:

In order to find
we consider the transpose
of the continuous induced map
defined by
As discussed above, for any
the transpose may be calculated by:
![{\displaystyle {\begin{aligned}\left\langle {}^{t}P(D_{f}),\phi \right\rangle &=\int _{U}f(x)P(\phi )(x)\,dx\\&=\int _{U}f(x)\left[\sum \nolimits _{\alpha }c_{\alpha }(x)(\partial ^{\alpha }\phi )(x)\right]\,dx\\&=\sum \nolimits _{\alpha }\int _{U}f(x)c_{\alpha }(x)(\partial ^{\alpha }\phi )(x)\,dx\\&=\sum \nolimits _{\alpha }(-1)^{|\alpha |}\int _{U}\phi (x)(\partial ^{\alpha }(c_{\alpha }f))(x)\,dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a73471cb9ab68b74797566d1588d6dcaf761940d)
For the last line we used integration by parts combined with the fact that
and therefore all the functions
have compact support.[note 17] Continuing the calculation above we have for all
![{\displaystyle {\begin{aligned}\left\langle {}^{t}P(D_{f}),\phi \right\rangle &=\sum \nolimits _{\alpha }(-1)^{|\alpha |}\int _{U}\phi (x)(\partial ^{\alpha }(c_{\alpha }f))(x)\,dx&&{\text{As shown above}}\\[4pt]&=\int _{U}\phi (x)\sum \nolimits _{\alpha }(-1)^{|\alpha |}(\partial ^{\alpha }(c_{\alpha }f))(x)\,dx\\[4pt]&=\int _{U}\phi (x)\sum _{\alpha }\left[\sum _{\gamma \leq \alpha }{\binom {\alpha }{\gamma }}(\partial ^{\gamma }c_{\alpha })(x)(\partial ^{\alpha -\gamma }f)(x)\right]\,dx&&{\text{Leibniz rule}}\\&=\int _{U}\phi (x)\left[\sum _{\alpha }\sum _{\gamma \leq \alpha }(-1)^{|\alpha |}{\binom {\alpha }{\gamma }}(\partial ^{\gamma }c_{\alpha })(x)(\partial ^{\alpha -\gamma }f)(x)\right]\,dx\\&=\int _{U}\phi (x)\left[\sum _{\alpha }\left[\sum _{\beta \geq \alpha }(-1)^{|\beta |}{\binom {\beta }{\alpha }}\left(\partial ^{\beta -\alpha }c_{\beta }\right)(x)\right](\partial ^{\alpha }f)(x)\right]\,dx&&{\text{Grouping terms by derivatives of }}f\\&=\int _{U}\phi (x)\left[\sum \nolimits _{\alpha }b_{\alpha }(x)(\partial ^{\alpha }f)(x)\right]\,dx&&b_{\alpha }:=\sum _{\beta \geq \alpha }(-1)^{|\beta |}{\binom {\beta }{\alpha }}\partial ^{\beta -\alpha }c_{\beta }\\&=\left\langle \left(\sum \nolimits _{\alpha }b_{\alpha }\partial ^{\alpha }\right)(f),\phi \right\rangle \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/135d1166231a190edfc094f39010b8432d361f32)
Define the formal transpose of
which will be denoted by
to avoid confusion with the transpose map, to be the following differential operator on U:

The computations above have shown that:
- Lemma. Let
be a linear differential operator with smooth coefficients in
Then for all
we have

- which is equivalent to:

The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, i.e.
enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator
defined by
We claim that the transpose of this map,
can be taken as
To see this, for every
, compute its action on a distribution of the form
with
:

We call the continuous linear operator
the differential operator on distributions extending P. Its action on an arbitrary distribution
is defined via:

If
converges to
then for every multi-index
converges to
Multiplication of distributions by smooth functions
A differential operator of order 0 is just multiplication by a smooth function. And conversely, if
is a smooth function then
is a differential operator of order 0, whose formal transpose is itself (i.e.
). The induced differential operator
maps a distribution T to a distribution denoted by
We have thus defined the multiplication of a distribution by a smooth function.
We now give an alternative presentation of multiplication by a smooth function. If
is a smooth function and T is a distribution on U, then the product mT is defined by

This definition coincides with the transpose definition since if
is the operator of multiplication by the function m (i.e.,
), then

so that
Under multiplication by smooth functions,
is a module over the ring
With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if ?? is the Dirac delta distribution on R, then m? = m(0)?, and if ?? is the derivative of the delta distribution, then

The bilinear multiplication map
given by
is not continuous; it is however, hypocontinuous.
Example. For any distribution T, the product of T with the function that is identically 1 on U is equal to T.
Example. Suppose
is a sequence of test functions on U that converges to the constant function
For any distribution T on U, the sequence
converges to 
If
converges to
and
converges to
then
converges to
Problem of multiplying distributions
It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if p.v. is the distribution obtained by the Cauchy principal value

If ? is the Dirac delta distribution then

but

so the product of a distribution by a smooth function (which is always well defined) cannot be extended to an associative product on the space of distributions.
Thus, nonlinear problems cannot be posed in general and thus not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) causal perturbation theory. This does not solve the problem in other situations. Many other interesting theories are non linear, like for example the Navier-Stokes equations of fluid dynamics.
Several not entirely satisfactory[] theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today.
Inspired by Lyons' rough path theory,[39]Martin Hairer proposed a consistent way of multiplying distributions with certain structure (regularity structures[40]), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli-Imkeller-Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.
Composition with a smooth function
Let T be a distribution on
Let V be an open set in
and F : V -> U. If F is a submersion, it is possible to define

This is the composition of the distribution T with F, and is also called the pullback of T along F, sometimes written

The pullback is often denoted F*, although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that F be a submersion is equivalent to the requirement that the Jacobian derivative dF(x) of F is a surjective linear map for every x ? V. A necessary (but not sufficient) condition for extending F# to distributions is that F be an open mapping.[41] The inverse function theorem ensures that a submersion satisfies this condition.
If F is a submersion, then F# is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since F# is a continuous linear operator on
Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.[42]
In the special case when F is a diffeomorphism from an open subset V of
onto an open subset U of
change of variables under the integral gives

In this particular case, then, F# is defined by the transpose formula:

Convolution
Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions.
Recall that if
and g are functions on
then we denote by
the convolution of
and g, defined at
to be the integral

provided that the integral exists. If
are such that 1/r = (1/p) + (1/q) - 1 then for any functions
and
we have
and
If
and g are continuous functions on
at least one of which has compact support, then
and if
then the value of
on A do not depend on the values of
outside of the Minkowski sum 
Importantly, if
has compact support then for any
the convolution map
is continuous when considered as the map
or as the map 
- Translation and symmetry
Given
the translation operator ?a sends
to
defined by
This can be extended by the transpose to distributions in the following way: given a distribution T, the translation of
by
is the distribution
defined by
[45]
Given
define the function
by
Given a distribution T, let
be the distribution defined by
The operator
is called the symmetry with respect to the origin.
Convolution of a test function with a distribution
Convolution with
defines a linear map:

which is continuous with respect to the canonical LF space topology on
Convolution of
with a distribution
can be defined by taking the transpose of Cf relative to the duality pairing of
with the space
of distributions. If
then by Fubini's theorem

Extending by continuity, the convolution of
with a distribution T is defined by

for all
An alternative way to define the convolution of a test function
and a distribution T is to use the translation operator ?a. The convolution of the compactly supported function
and the distribution T is then the function defined for each
by

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution T has compact support then if
is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on
to
the restriction of an entire function of exponential type in
to
) then the same is true of
If the distribution T has compact support as well, then
is a compactly supported function, and the Titchmarsh convolution theorem Hörmander (1983, Theorem 4.3.3) implies that

where ch denotes the convex hull and supp denotes the support.
Convolution of a smooth function with a distribution
Let
and
and assume that at least one of
and T has compact support. The convolution of
and T, denoted by
or by
is the smooth function:

satisfying for all
:
![{\displaystyle {\begin{aligned}&\operatorname {supp} (f\ast T)\subseteq \operatorname {supp} (f)+\operatorname {supp} (T)\\[6pt]&{\text{ for all }}p\in \mathbb {N} ^{n}:\quad {\begin{cases}\partial ^{p}\left\langle T,\tau _{x}{\tilde {f}}\right\rangle =\left\langle T,\partial ^{p}\tau _{x}{\tilde {f}}\right\rangle \\\partial ^{p}(T\ast f)=(\partial ^{p}T)\ast f=T\ast (\partial ^{p}f)\end{cases}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09b5e643a45fd5be500ccecd71f95c2a23d325)
If T is a distribution then the map
is continuous as a map
where if in addition T has compact support then it is also continuous as the map
and continuous as the map 
If
is a continuous linear map such that
for all
and all
then there exists a distribution
such that
for all 
Example. Let H be the Heaviside function on R. For any

Let
be the Dirac measure at 0 and
its derivative as a distribution. Then
and
Importantly, the associative law fails to hold:

Convolution of distributions
It is also possible to define the convolution of two distributions S and T on
provided one of them has compact support. Informally, in order to define S * T where T has compact support, the idea is to extend the definition of the convolution * to a linear operation on distributions so that the associativity formula

continues to hold for all test functions
[47]
It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that S and T are distributions and that S has compact support. Then the linear maps

are continuous. The transposes of these maps,

are consequently continuous and one may show that

This common value is called the convolution of S and T and it is a distribution that is denoted by
or
It satisfies
If S and T are two distributions, at least one of which has compact support, then for any
If T is a distribution in
and if
is a Dirac measure then 
Suppose that it is T that has compact support. For
consider the function

It can be readily shown that this defines a smooth function of x, which moreover has compact support. The convolution of S and T is defined by

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index ?,

The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.
This definition of convolution remains valid under less restrictive assumptions about S and T.[48]
The convolution of distributions with compact support induces a continuous bilinear map
defined by
where
denotes the space of distributions with compact support. However, the convolution map as a function
is not continuous although it is separately continuous. The convolution maps
and
given by
both fail to be continuous. Each of these non-continuous maps is, however, separately continuous and hypocontinuous.
Convolution versus multiplication
In general, regularity is required for multiplication products and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let
be a rapidly decreasing tempered distribution or, equivalently,
be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let
be the normalized (unitary, ordinary frequency) Fourier transform [50] then, according to Schwartz (1951),


hold within the space of tempered distributions.[51][52][53] In particular, these equations become the Poisson Summation Formula if
is the Dirac Comb.[54] The space of all rapidly decreasing tempered distributions is also called the space of convolution operators
and the space of all ordinary functions within the space of tempered distributions is also called the space of multiplication operators
More generally,
and
[56] A particular case is the Paley-Wiener-Schwartz Theorem which states that
and
This is because
and
In other words, compactly supported tempered distributions
belong to the space of convolution operators
and
Paley-Wiener functions
better known as bandlimited functions, belong to the space of multiplication operators 
For example, let
be the Dirac comb and
be the Dirac delta then
is the function that is constantly one and both equations yield the Dirac comb identity. Another example is to let
be the Dirac comb and
be the rectangular function then
is the sinc function and both equations yield the Classical Sampling Theorem for suitable
functions. More generally, if
is the Dirac comb and
is a smooth window function (Schwartz function), e.g. the Gaussian, then
is another smooth window function (Schwartz function). They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.
Tensor product of distributions
Let
and
be open sets. Assume all vector spaces to be over the field
where
or
For
we define the following family of functions:

Given
and
we define the following functions:
![{\displaystyle {\begin{aligned}{\begin{cases}\langle S,f^{\bullet }\rangle :V\to \mathbb {F} \\y\mapsto \langle S,f^{y}\rangle \end{cases}}\\[8pt]{\begin{cases}\langle T,f_{\bullet }\rangle :U\to \mathbb {F} \\x\mapsto \langle T,f_{x}\rangle \end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf574ef951ef30d4d9c547662c7316dbb7fb304)
Note that
and
Now we define the following continuous linear maps associated to
and
:
![{\displaystyle {\begin{aligned}{\mathcal {D}}'(U)\ni S&\longrightarrow {\begin{cases}{\mathcal {D}}(U\times V)\to {\mathcal {D}}(V)\\f\mapsto \langle S,f^{\bullet }\rangle \end{cases}}\\[8pt]{\mathcal {D}}'(V)\ni T&\longrightarrow {\begin{cases}{\mathcal {D}}(U\times V)\to {\mathcal {D}}(U)\\f\mapsto \langle T,f_{\bullet }\rangle \end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cea2abcad076a24cdf12500a7430d08fb7095ad8)
Moreover if either
(resp.
) has compact support then it also induces a continuous linear map of
(resp.
).
Definition. The tensor product of
and
denoted by
or
is a distribution in
and is 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.
Schwartz kernel theorem — We have canonical TVS isomorphisms:

Here
represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product, since these spaces are nuclear) and
has the topology of uniform convergence on bounded subsets.
This result does not hold for Hilbert spaces 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.
Spaces of distributions
For all 0 < k < ? and all 1 < p < ?, all of the following canonical injections are continuous and have a range that is dense in their codomain:

where the topologies on
(
) are defined as direct limits of the spaces
in a manner analogous to how the topologies on
were defined (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 every
All of the canonical injections
(
) are continuous and the range of this injection is dense in the codomain if and only if
(here
has its usual norm topology).
Suppose that
is one of the spaces
(
) or
(
) or
(
). Since the canonical injection
is a continuous injection whose image is dense in the codomain, the transpose
is a continuous injection. This transpose thus allows us to identify
with a certain vector subspace of the space of distributions. This transpose map is not necessarily a TVS-embedding so that topology that this map transfers 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 a space of distributions.
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 X may, through the transpose
be transferred directly to elements of the space
Radon measures
The natural inclusion
is a continuous injection whose image is dense in its codomain, so the transpose
is also a continuous injection.
Note that the continuous dual space
can be identified as the space of Radon measures, 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 U such that for all
and
- if
is a Radon measure on U then the linear functional on
defined by
is continuous.
Through the injection
every Radon measure becomes a distribution on U. If
is a locally integrable function on U 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 measures, which shows that every Radon measure can be written as a sum of derivatives of locally
functions in U:
- Theorem. Suppose
is a Radon measure, V ? U is a neighborhood of the support of T, and
There exists is a family of locally
functions in U such that

- and for very

- Positive Radon measures
A linear function T on a space of functions is called positive if whenever a function
that belongs to the domain of T is non-negative (i.e.
is real-valued and
) then
One may show that every positive linear functional on
is necessarily continuous (i.e. necessarily a Radon measure).Note that Lebesgue measure 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 locally integrable if it is Lebesgue integrable over every compact subset K of U.[note 18] This is a large class of functions which includes all continuous functions and all Lp 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 Tf, whose value on the test function
is given by the Lebesgue integral:

Conventionally, one abuses notation by identifying Tf with
provided no confusion can arise, and thus the pairing between Tf and
is often written

If
and g are two locally integrable functions, then the associated distributions Tf and Tg are equal to the same element of