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 where 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 singular, such as the Dirac delta function.
A function is normally thought of as acting on the points in the function domain by "sending" a point 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.
In applications to physics and engineering, test functions are usually infinitely differentiablecomplex-valued (or real-valued) functions with compactsupport that are defined on some given non-empty open subset. (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by or
Most commonly encountered functions, including all continuous maps if using can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that "acts on" a test function by "sending" it to the number which is often denoted by This new action of is a scalar-valued map, denoted by whose domain is the space of test functions This functional turns out to have the two defining properties of what is known as a distribution on : it is linear and also continuous when is given a certain topology called the canonical LF topology. The action of this distribution on a test function can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like that arise from functions in this way are prototypical examples of distributions, but many cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions against certain measures. It is nonetheless still possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.
More generally, a distribution on 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 , 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.
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.
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:
Definitions of test functions and distributions
In this section, some basic notions and definitions needed to define real-valued distributions on U are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on spaces of test functions and distributions.
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.
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 all compact subsets of
If is a real-valued function on , 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 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.
For all and any compact subsets and of , we have:
Definition: Elements of are called test functions on U and is called the space of test functions on U. We will use both and to denote this space.
Distributions on U are continuous linear functionals on when this vector space is endowed with a particular topology called the canonical LF-topology. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on that are often straightforward to verify.
Proposition: A linear functionalT on is continuous, and therefore a distribution, if and only if either of the following equivalent conditions is satisfied:
For every compact subset there exist constants and (dependent on ) such that for all with support contained in ,
For every compact subset and every sequence in whose supports are contained in , if converges uniformly to zero on for every multi-index, then
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.
Suppose and is an arbitrary compact subset of Suppose an integer such that [note 1] and is a multi-index with length For define:
while for define all the functions above to be the constant 0 map.
generate the same locally convexvector topology on (so for example, the topology generated by the seminorms in is equal to the topology generated by those in ).
The vector space is endowed with the locally convex topology induced by any one of the four families of seminorms described above. This topology is also equal to the vector topology induced by all of the seminorms in
Trivial extensions and independence of Ck(K)'s topology from U
Suppose is an open subset of and is a compact subset. By definition, elements of are functions with domain (in symbols, ), so the space and its topology depend on to make this dependence on the open set clear, temporarily denote by
Importantly, changing the set to a different open subset (with ) will change the set from to [note 3] so that elements of will be functions with domain instead of
Despite depending on the open set (), the standard notation for makes no mention of it.
This is justified because, as this subsection will now explain, the space is canonically identified as a subspace of (both algebraically and topologically).
It is enough to explain how to canonically identify with when one of and is a subset of the other. The reason is that if and are arbitrary open subsets of containing then the open set also contains so that each of and is canonically identified with and now by transitivity, is thus identified with
So assume are open subsets of containing
Given its trivial extension to is the function defined by:
This trivial extension belongs to (because has compact support) and it will be denoted by (that is, ). The assignment thus induces a map that sends a function in to its trivial extension on This map is a linear injection and for every compact subset (where is also a compact subset of since ),
the vector space is canonically identified with its image in Because through this identification, can also be considered as a subset of
Thus the topology on is independent of the open subset of that contains  which justifies the practice of writing instead of
Recall that denotes all functions in that have compact support in where note that is the union of all as ranges over all compact subsets of Moreover, for each 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 Unless indicated otherwise, it is endowed with a topology called the canonical LF topology, whose definition is given in the article: Spaces of test functions and distributions.
explicitly, for every Mackey convergent null sequence in the sequence is bounded;
a sequence is said to be Mackey convergent to the origin if there exists a divergent sequence of positive real numbers such that the sequence is bounded; every sequence that is Mackey convergent to the origin 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 on such that
There exists a constant and a finite subset (where is any collection of continuous seminorms that defines the canonical LF topology on ) such that [note 6]
For every compact subset there exist constants and such that for all 
For every compact subset there exist constants and such that for all with support contained in 
For any compact subset and any sequence in if converges uniformly to zero for all multi-indices then
Topology on the space of distributions and its relation to the weak-* topology
Neither nor its strong dual is a sequential space and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is not enough to fully/correctly define their topologies).
However, a sequence in converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to define the convergence of a sequence of distributions; this is fine for sequences but this is not guaranteed to extend to the convergence of nets of distributions because a net may converge pointwise but fail to converge in the strong dual topology).
More information about the topology that is endowed with can be found in the article on spaces of test functions and distributions and the articles on polar topologies and dual systems.
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.
Extensions and restrictions to an open subset
Let be open subsets of
Every function can be extended by zero from its domain V to a function on U by setting it equal to on the complement This extension is a smooth compactly supported function called the trivial extension of to and it will be denoted by
This assignment defines the trivial extension operator
which is a continuous injective linear map. It is used to canonically identify as a vector subspace of (although not as a topological subspace).
Its transpose (explained here)
is called the restriction to of distributions in  and as the name suggests, the image of a distribution under this map is a distribution on called the restriction of to The defining condition of the restriction is:
If then the (continuous injective linear) trivial extension map is not a topological embedding (in other words, if this linear injection was used to identify as a subset of then 's topology would strictly finer than the subspace topology that induces on it; importantly, it would not be a topological subspace since that requires equality of topologies) and its range is also not dense in its codomain  Consequently if then the restriction mapping is neither injective nor surjective. 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 
Unless 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 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
Corollary — Let be a collection of open subsets of and let 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.
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 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 If the support of a distribution T is compact then it has finite order and there is a constant and a non-negative integer such that:
If T has compact support, then it has a unique extension to a continuous linear functional on ; this function 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  Furthermore, if is a differential operator in U, then for all distributions T on U and all we have and 
For any let denote the distribution induced by the Dirac measure at 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 such that
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 Uwith support in V such that
where the derivatives are understood in the sense of distributions. That is, for all test functions on U,
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.
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 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 on U such that
any compact subset K of U intersects the support of only finitely many and
Moreover, if T has finite order, then one can choose 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 that intersect the support of
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if is a linear map that is continuous with respect to the weak topology, then it is possible to extend to a map by passing to the limit.[note 7][clarification needed]
Preliminaries: Transpose of a linear operator
Operations on distributions and spaces of distributions are often defined using the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis. For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space). In general, the transpose of a continuous linear map is the linear map
or equivalently, it is the unique map satisfying for all and all (the prime symbol in does not denote a derivative of any kind; it merely indicates that is an element of the continuous dual space ). Since is continuous, the transpose is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).
In the context of distributions, the characterization of the transpose can be refined slightly. Let be a continuous linear map. Then by definition, the transpose of is the unique linear operator that satisfies:
Since is dense in (here, actually refers to the set of distributions ) it is sufficient that the defining equality hold for all distributions of the form where Explicitly, this means that a continuous linear map is equal to if and only if the condition below holds:
Let be the partial derivative operator To extend we compute its transpose:
Therefore Thus, 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 is a distribution in then
where is the derivative of and is a translation by thus the derivative of may be viewed as a limit of quotients.
Differential operators acting on smooth functions
A linear differential operator in with smooth coefficients acts on the space of smooth functions on Given such an operator
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:
With this notation, the diagram commuting is equivalent to:
To find the transpose of the continuous induced map defined by is considered in the lemma below.
This leads to the following definition of the differential operator on called the formal transpose of which will be denoted by to avoid confusion with the transpose map, that is defined by
Lemma — Let be a linear differential operator with smooth coefficients in Then for all we have
which is equivalent to:
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 8] Continuing the calculation above, for all
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is,  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 . 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 (that is, ). The induced differential operator maps a distribution 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 the multiplication of a distribution on by a smooth function The product is defined by
This definition coincides with the transpose definition since if is the operator of multiplication by the function (that is, ), 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, some unusual identities also arise. For example, if is the Dirac delta distribution on then and if is the derivative of the delta distribution, then
The bilinear multiplication map given by is not continuous; it is however, hypocontinuous.
Example. The product of any distribution with the function that is identically 1 on is equal to
Example. Suppose is a sequence of test functions on that converges to the constant function For any distribution on the sequence converges to 
If converges to and converges to then converges to
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 is the distribution obtained by the Cauchy principal value
If is the Dirac delta distribution then
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.
Inspired by Lyons' rough path theory,Martin Hairer proposed a consistent way of multiplying distributions with certain structures (regularity structures), 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.
Let be a distribution on Let be an open set in and If is a submersion then it is possible to define
This is the composition of the distribution with , and is also called the pullback of along , sometimes written
The pullback is often denoted although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that be a submersion is equivalent to the requirement that the Jacobian derivative of is a surjective linear map for every A necessary (but not sufficient) condition for extending to distributions is that be an open mapping. The Inverse function theorem ensures that a submersion satisfies this condition.
If is a submersion, then is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since 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.
In the special case when is a diffeomorphism from an open subset of onto an open subset of change of variables under the integral gives:
In this particular case, then, is defined by the transpose formula:
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 are functions on then we denote by the convolution of and defined at to be the integral
provided that the integral exists. If are such that then for any functions and we have and  If and are continuous functions on at least one of which has compact support, then and if then the value of on 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 
Given the translation operator sends to defined by This can be extended by the transpose to distributions in the following way: given a distribution the translation of by is the distribution defined by 
Given define the function by Given a distribution 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 of with a distribution can be defined by taking the transpose of 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 is defined by
An alternative way to define the convolution of a test function and a distribution is to use the translation operator The convolution of the compactly supported function and the distribution 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 has compact support, and 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 has compact support as well, then is a compactly supported function, and the Titchmarsh convolution theoremHörmander (1983, Theorem 4.3.3) implies that:
It is also possible to define the convolution of two distributions and on provided one of them has compact support. Informally, to define where has compact support, the idea is to extend the definition of the convolution