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Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak 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 the 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 such a function "acts on" a test function by "sending" it to the number which is often denoted by This new action of defines a scalar-valued map 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 it is also continuous when is given a certain topology called the canonical LF topology. The action (the integration ) 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 there exist many distributions that 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 on Nonetheless, it is still always 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's 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-orderhyperbolic 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.
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:
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
If then the domain of is U and not K. So although depends on both K and U, only K is typically indicated. The justification for this common practice is detailed below. The notation will only be used 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 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 any 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 ,[1][2]
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 is an integer such that [note 1] and is a multi-index with length For and define:
while for define all the functions above to be the constant 0 map.
Each of the following sets of seminorms 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
With this topology, becomes a locally convex Fréchet space that is notnormable. Every element of is a continuous seminorm on Under this topology, a net in converges to if and only if for every multi-index with and every compact the net of partial derivatives converges uniformly to on [3] For any any (von Neumann) bounded subset of is a relatively compact subset of [4] In particular, a subset of is bounded if and only if it is bounded in for all [4] The space is a Montel space if and only if [5]
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
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 ), If is restricted to then the following induced linear map is a homeomorphism (linear homeomorphisms are called TVS-isomorphisms): and thus the next map is a topological embedding: Using the injection 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 [7] 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.
As discussed earlier, continuous linear functionals on a are known as distributions on Other equivalent definitions are described below.
By definition, a distribution on is a continuouslinear functional on Said differently, a distribution on is an element of the continuous dual space of when is endowed with its canonical LF topology.
There is a canonical duality pairing between a distribution on and a test function which is denoted using angle brackets by
One interprets this notation as the distribution acting on the test function to give a scalar, or symmetrically as the test function acting on the distribution
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 [1]
For every compact subset there exist constants and such that for all with support contained in [10]
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
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 indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes into a completenuclear 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 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.
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 [11] 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[11] Consequently if then the restriction mapping is neither injective nor surjective.[11] 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 [11]
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
Theorem[12] — 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[12] — 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[12] — 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.[12] 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.[12] If is continuous, then the support of is equal to the closure of the set of points in U at which does not vanish.[12] The support of the distribution associated with the Dirac measure at a point