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
engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the
Dirac delta function.
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
complex-valued (or sometimes
real-valued) functions with
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
[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
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.
The following notation will be used throughout this article:
- is a fixed positive integer and is a fixed non-empty
open subset of
- denotes the
- 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
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 (
- Let denote the
vector space of all k-times
continuously differentiable real-valued functions on U.
- For any compact subset let and both denote the vector space of all those functions such that
- Note that depends on both K and U but we will only indicate K, where in particular, if then the domain of is U rather than K. We will use the notation only when the notation risks being ambiguous.
- Clearly, every contains the constant 0 map, even if
- Let denote the set of all such that for some compact subset K of U.
- Equivalently, is the set of all such that has compact support.
- is equal to the union of all as ranges over
- If is a real-valued function on U, then is an element of if and only if is a
bump function. Every real-valued test function on is always also a complex-valued test function on
The graph of the
This function is a test function on
and is an element of
The support of this function is the closed
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:
: Elements of
are called test functions
is called the space of test function
. We will use both
to denote this space.
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 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
- 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. First, a (
non-normable) topology on will be defined, then every will be endowed with the
subspace topology 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 of is then endowed with the (non-metrizable)
strong dual topology 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
This finally permits consideration of more advanced notions such as convergence of distributions (both sequences and nets), various (sub)spaces of distributions, and operations on distributions, including extending differential equations to distributions.
- Choice of compact sets
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 i, and is a
relatively compact non-empty open subset of (here, "relatively compact" means that the
closure of in either U or is compact).
We make into a
directed set 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 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.
is an arbitrary compact subset of
an integer such that
is a multi-index with length
we define all the functions above to be the constant 0
Each of the functions above are non-negative -valued
Each of the following families of seminorms generates the same
vector topology on :
: We will henceforth assume that
is endowed with the
topology defined by any (or equivalently, all) of the families of
With this topology, becomes a locally convex (
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 with and every the net of partial derivatives
converges uniformly 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 i 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 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 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 Ck(K)
As before, fix Recall that if is any compact subset of then
: For any compact subset
we will henceforth assume that
is endowed with the
it inherits from the
For any compact subset is a closed subspace of the Fréchet space and is thus also a
Fréchet space. For all compact satisfying denote the
inclusion map 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 (or "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
And when then is even a
Hilbert space. The space is a
Montel space so if then it is not
normable and thus not a Banach space (although like all other it is a
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 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 (since is also a compact subset of ). 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
(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
Canonical LF topology
Recall that denote all those functions in that have compact support in where note that is the union of all as K ranges over Moreover, for every k, is a dense subset of The special case when gives us the space of test functions.
is called the space of test functions on
and it may also be denoted by
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 K and L, we declare that if and only if which in particular makes the collection of compact subsets of U into a
directed set (we say that such a collection is directed by subset inclusion). For all compact satisfying there are
Recall from above that the map is a
topological embedding. The collection of maps
direct system in the
locally convex topological vector spaces that is
directed 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 locally convex topology making all of the inclusion maps continuous.
The canonical LF topology on
making all of the inclusion maps
continuous (where K
: As is common in mathematics literature, this article will henceforth assume that
is endowed with its canonical LF topology (unless explicitly stated otherwise).
- Topology defined by 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:
is a neighborhood of the origin in
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
- 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 the canonical LF topology on is the weakest locally convex TVS topology making all linear differential operators in U 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 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 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
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).
- 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 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) then is not a dense subset of and is not a
topological embedding. Consequently, if 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 such that and B is a bounded subset of Moreover, if is compact and then S 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
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 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 though it does not admit a metric). The fact that is a
Montel space makes up for the non-metrizability of (see this footnote for a more detailed explanation).
- Relationships between spaces
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
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
- Other properties
- The differentiation map is a surjective continuous linear operator.
bilinear multiplication map given by is not continuous; it is however,
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
By definition, a distribution on U
is defined to be a
Said differently, a distribution on U
is an element of the
continuous dual space
is endowed with its canonical LF topology.
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
- T is
continuous at the origin;
- T is
- T is a
- T is
- explicitly, for every sequence in that converges in to some
- 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 is said to be
Mackey convergent to 0 if there exists a divergent sequence of positive real number such that the sequence 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 closed;
- There exists a continuous seminorm g 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
- 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 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
Topology on the space of distributions
Definition and notation
: The space of distributions on U
, denoted by
continuous dual space
endowed with the
topology of uniform convergence on bounded subsets
More succinctly, the space of distributions on U
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
Montel space and it is with this topology that the
kernels theorem of Schwartz holds.
 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-
topological vector space. The space is
 and has the strong
 but it is neither a
 nor a
 which in particular implies that it is not
metrizable and also that its topology can not be defined using only sequences.
- Topological vector space categories
The canonical LF topology makes into a
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, is a complete
Mackey space. The
strong dual of is a
Fréchet space if and only if 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
bornological spaces, which implies that all six of these locally convex spaces are also
Mackey spaces. The spaces and are both
Fréchet spaces. Moreover, both and are
- Convergent sequences and their insufficiency to describe topologies
The strong dual spaces of and are
sequential spaces but not
 Moreover, neither the space of test functions nor its strong dual is a
sequential space (not even an
 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 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
Neither the space nor its strong dual is a
 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 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 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 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
and is the distribution corresponding to then
as so δ 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 value at (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
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
Restrictions to an open subset
Let U and V be open subsets of with . 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 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 ρVU(T) is a distribution in defined by:
Unless U = V, the restriction to V is neither
surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if and then the distribution
is in but admits no extension to
Gluing and distributions that vanish in a set
Theorem — Let be a collection of open subsets of For each let and suppose that for all the restriction of to is equal to the restriction of to (note that both restrictions are elements of ). Then there exists a unique such that for all the restriction of T to is equal to
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 ρVU.
- 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 δ function at P. That is, there exists an integer m and complex constants aα such that
where is the translation operator.
- Distribution with compact support
Theorem — Suppose T is a distribution on U with compact support K. There exists a continuous function defined on U and a multi-index p such that
where the derivatives are understood in the sense of distributions. That is, for all test functions on U,
- 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
Theorem — Let T be a distribution on U.
There exists a sequence in such that each Ti has compact support and every compact subset intersects the support of only finitely many Ti, and the sequence of partial sums defined by converges in to T; in other words we have:
Recall that a sequence converges in (with its strong dual topology) if and only if it converges pointwise.
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 gα 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.
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.
 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
Differentiation of distributions
Let is the partial derivative operator In order to extend we compute its transpose:
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 then
where is the derivative of T and τx 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:
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
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
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 δ′ is the Dirac delta distribution on , then mδ = m(0)δ, and if δ′ is the derivative of the delta distribution, then
The bilinear multiplication map given by is not continuous; it is however,
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.