This article may be too long to read and navigate comfortably. (December 2020)
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.
The formal definitions of the spaces of test functions and distributions, their topologies, and their properties is highly technical and so a full presentation of this is given in the article on spaces of test functions and distributions. Whereas that article is primarily concerned with spaces of test functions and distributions, this article is primarily concerned with individual test functions and distributions.
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 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:
Definitions of test functions and distributions
In this section, some basic notions and defintions needed to define real-valued distributions on U are introduced. The formal definitions of the spaces of test functions and distributions, their topologies, and their properties is highly technical and so a full presentation of this is given in the article on spaces of test functions and distributions. Content related to the formal definition of these spaces that is also used in this article is described in this subsection.
- 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 all compact subsets of
- 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
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 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
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.
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 with and every compact 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 
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
Topology on Ck(K)
As before, fix Recall that if is any compact subset of then
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:
Canonical LF topology
Recall that denote all those functions in that have compact support in where note that is the union of all as ranges over all compact subsets of Moreover, for every is a dense subset of The special case when gives us the space of test functions.
The canonical LF-topology is not metrizable and importantly, it is strictly finer than the subspace topology that induces on However, the canonical LF-topology does make into a complete reflexive nuclear Montel bornological barrelled Mackey space; the same is true of its strong dual space (i.e. the space of all distributions with its usual topology). The canonical LF-topology can be defined in various ways.
As discussed earlier, continuous linear functionals on a are known as distributions on Other equivalent definitions are described below.
One interprets this notation as the distribution acting on the test function to give a scalar, or symmetrically as the test function acting on the distribution
Topology on the space of distributions and its relation to the weak-* topology
The set of all distributions on is the continuous dual space of which when endowed with the strong dual topology is denoted by Importantly, unless indicated otherwise, the topology on is the strong dual topology; if the topology is instead the weak-* topology then this will be clearly indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes into a complete nuclear space, to name just a few of its desirable properties.
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 actually 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 in the articles on polar topologies and dual systems.
A linear map from into another locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general topological spaces (e.g. that are not also locally convex topological vector spaces). The same is true of maps from (more generally, this is true of maps from any locally convex bornological space).
Characterizations of distributions
Proposition. If 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 4]
- T is sequentially continuous at the origin; in other words, T maps null sequences[note 5] 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 convergent null sequences 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 [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 p, then
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.
Extensions and 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. 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 is a distribution in defined by:
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.
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 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 furthermore, 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 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  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 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
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
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
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
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:
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:
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
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 7][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. 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:
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:
Differentiation of distributions
Let be 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
Differential operators acting on smooth functions
A linear differential operator in U 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:
In order 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
As discussed above, for any the transpose may be calculated by:
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 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 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 is the Dirac delta distribution on then and if is the derivative of the delta distribution, then
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 is the distribution obtained by the Cauchy principal value
If is the Dirac delta distribution then
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, Martin Hairer proposed a consistent way of multiplying distributions with certain structure (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.
Composition with a smooth function
Let T be a distribution on Let V be an open set in and 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 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 of F is a surjective linear map for every A necessary (but not sufficient) condition for extending to distributions is that F be an open mapping. The Inverse function theorem ensures that a submersion satisfies this condition.
If F is a submersion, then is defined on distributions by finding the transpose map. 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 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, 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
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 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 
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:
Extending by continuity, the convolution of with a distribution T is defined by
An alternative way to define the convolution of a test function and a distribution T is to use the translation operator 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
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:
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 
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 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
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
This common value is called the convolution of S and Tand 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 ; thus is the identity element of the convolution operation. Moreover, if is a function then where now the associativity of convolution implies that for all functions and
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 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
This definition of convolution remains valid under less restrictive assumptions about S and T.
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 then, according to Schwartz (1951),
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 define for every and every the following functions:
Given and define the following functions:
Moreover if either (resp. ) has compact support then it also induces a continuous linear map of