Spaces of test functions and distributions


In mathematical analysis, the spaces of test functions and distributions are topological vector spaces that are used in the definition and application of distributions.
Test functions are usually infinitely differentiable complex-valued functions on a non-empty open subset that have compact support.
The space of all test functions, denoted by, is endowed with a certain topology, called the, that makes into a complete Hausdorff [Locally Convex set|convex topological vector space|locally convex] TVS.
The dual space of is called and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology.
There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If then the use of Schwartz functions as test functions gives rise to a certain subspace of whose elements are called . 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 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]. Use of analytic test functions leads to Sato's theory of hyperfunctions.

Notation

The following notation will be used throughout this article:
  • is a fixed positive integer and is a fixed non-empty open subset of Euclidean space
  • denotes the natural numbers.
  • will denote a non-negative integer or
  • If is a function then will denote its domain and the ' of denoted by is defined to be the closure of the set in
  • For two functions, the following notation defines a canonical pairing:
  • A of size is an element in . The ' 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 .

Definitions of test functions and distributions

In this section, we will formally define real-valued distributions on. With minor modifications, one can also define complex-valued distributions, and one can replace with any smooth manifold.
Note that for all and any compact subsets and of, we have:
Distributions on are defined to be the continuous linear functionals on when this vector space is endowed with a particular topology called the .
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 is a linear functional on then the is a distribution if and only if the following equivalent conditions are satisfied:
  1. For every compact subset there exist constants and such that for all
  2. For every compact subset there exist constants and such that for all with support contained in
  3. 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 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 locally convex topological vector spaces be defined first. First, a topology on will be defined, then every will be endowed with the subspace topology induced on it by, and finally the canonical LF-topology on will be defined.
The space of distributions, being defined as the continuous dual space of, is then endowed with the strong dual topology induced by and the canonical LF-topology.
This finally permits consideration of more advanced notions such as convergence of distributions, various spaces of distributions, and operations on distributions, including extending differential equations to distributions.

Choice of compact sets K

Throughout, will be any collection of compact subsets of such that and for any compact there exists some such that The most common choices for are:
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 ''C''''k''(''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.
All of the functions above are non-negative -valued seminorms on As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.
Each of the following sets of seminorms
generate the same locally convex vector topology on .
With this topology, becomes a locally convex Fréchet space that is normable. 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 For any any [Bounded set (topological vector space)|(von Neumann) Bounded linear map|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 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 defining families of seminorms. For example, using the seminorms results in the metric
Often, it is easier to just consider seminorms and use the tools of functional analysis.

Topology on ''C''''k''(''K'')

As before, fix Recall that if is any compact subset of then
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 whose image is closed in its codomain; said differently, the topology on is identical to the subspace topology it inherits from and also is a closed subset of The interior of relative to is empty.
If is finite then is a Banach space with a topology that can be defined by the norm
And when then is even a Hilbert space. The space is a distinguished Schwartz Montel space so if then it is normable and thus a Banach space.

Trivial extensions and independence of ''C''''k''(''K'')'s topology from ''U''

The definition of depends on so we will let denote the topological space which by definition is a topological subspace of Suppose is an open subset of containing and for any compact subset, let is the vector subspace of consisting of maps with support contained in Given, its is by definition, the function defined by:
so that Let denote the map that sends a function in to its trivial extension on. This map is a linear injection and for every compact subset we have
If is restricted to then the following induced linear map is a homeomorphism :
and thus the next two maps are topological embeddings:
.
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 Importantly, the subspace topology inherits from is identical to the subspace topology that it inherits from . Thus the topology on is independent of the open subset of that contains. This justifies the practice of writing 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 ranges over Moreover, for every, is a dense subset of The special case when gives us the space of test functions.
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 and, we declare that if and only if, which in particular makes the collection of compact subsets of into a directed set. For all compact satisfying there are inclusion maps
Recall from above that the map is a topological embedding. The collection of maps
forms a direct system in the category of locally convex topological vector spaces that is directed by . This system's direct limit is the pair where are the natural inclusions and where is now endowed with the strongest locally convex topology making all of the inclusion maps continuous.

Topology defined by neighborhoods of the origin

If is a convex subset of, then is a neighborhood of the origin in the canonical LF topology if and only if it satisfies the following condition:
Note that any convex set satisfying this condition is necessarily absorbing in. Since the topology of any topological vector space is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually the canonical LF topology by declaring that a convex balanced subset is a neighborhood of the origin if and only if it satisfies condition.

Topology defined via differential operators

A is a sum
where and all but finitely many of are identically. The integer is called the of the differential operator If is a linear differential operator of order 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 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, we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes into a Hausdorff locally convex strict LF-space, which of course is the reason why this topology is called "the canonical LF topology".
Universal property
From the universal property of direct limits, we know that if is a linear map into a locally convex space, then is continuous if and only if is bounded if and only if for every, the restriction of to is continuous.
Dependence of the canonical LF topology on
Suppose is an open subset of containing Let denote the map that sends a function in to its trivial extension on . This map is a continuous linear map. If then is a dense subset of and is a topological embedding. Consequently, if then the transpose of is neither one-to-one nor onto.
Bounded subsets
A subset is bounded in if and only if there exists some such that and is a bounded subset of Moreover, if is compact and then is bounded in if and only if it is bounded in For any any bounded subset of is a relatively compact subset of, where.
Non-metrizability
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 metrizable and thus also normable. The fact that is a nuclear Montel space makes up for the non-metrizability of .
Relationships between spaces
Using the universal property of direct limits and the fact that the natural inclusions are all topological embedding, one may show that all of the maps are also topological embeddings. Said differently, the topology on is identical to the subspace topology that it inherits from, where recall that 's topology was to be the subspace topology induced on it by. In particular, both and induces the same subspace topology on. However, this does ' imply that the canonical LF topology on is equal to the subspace topology induced on by ; these two topologies on are in fact ' equal to each other since the canonical LF topology is metrizable while the subspace topology induced on it by is metrizable. The canonical LF topology on is actually than the subspace topology that it inherits from .
Indeed, the canonical LF topology is so fine that if denotes some linear map that is a "natural inclusion" then this map will typically be continuous, which is ultimately the reason why locally integrable functions, Radon measures, etc. all induce distributions. 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.
Other properties

Distributions

As discussed earlier, continuous linear functionals on a are known as distributions on. Thus the set of all distributions on is the continuous dual space of, which when endowed with the strong dual topology is denoted by.
We have the canonical duality pairing between a distribution 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.

Characterizations of distributions

Proposition. If is a linear functional on then the following are equivalent:
  1. is a distribution;
  2. : is a continuous function.
  3. is continuous at the origin.
  4. is uniformly continuous.
  5. is a bounded operator.
  6. is sequentially continuous.
  7. * explicitly, for every sequence in that converges in to some
  8. is sequentially continuous at the origin; in other words, maps null sequences to null sequences.
  9. * explicitly, for every sequence in that converges in to the origin,
  10. * a is by definition a sequence that converges to the origin.
  11. maps null sequences to bounded subsets.
  12. * explicitly, for every sequence in that converges in to the origin, the sequence is bounded.
  13. maps Mackey convergent null sequences to bounded subsets;
  14. * explicitly, for every Mackey convergent null sequence in, the sequence is bounded.
  15. * a sequence is said to be if there exists a divergent sequence of positive real number such that the sequence is bounded; every sequence that is Mackey convergent to necessarily converges to the origin.
  16. The kernel of is a closed subspace of.
  17. The graph of is closed.
  18. There exists a continuous seminorm on such that
  19. There exists a constant a collection of continuous seminorms, that defines the canonical LF topology of and a finite subset such that
  20. For every compact subset there exist constants and such that for all
  21. For every compact subset there exist constants and such that for all with support contained in
  22. For any compact subset and any sequence in if converges uniformly to zero for all multi-indices then
  23. Any of the statements immediately above but with the additional requirement that compact set belongs to

Topology on the space of distributions

The topology of uniform convergence on bounded subsets is also called. This topology is chosen because it is with this topology that becomes a nuclear Montel space and it is with this topology that the Schwartz kernel theorem holds. No matter what dual topology is placed on a of distributions converges in this topology if and only if it converges pointwise. No matter which topology is chosen, will be a non-metrizable, locally convex topological vector space. The space is separable and has the strong Pytkeev property but it is neither a k-space nor a sequential space, which in particular implies that it is not metrizable and also that its topology can be defined using only sequences.

Topological properties

Topological vector space categories

The canonical LF topology makes into a complete distinguished strict LF-space, which implies that is a meager subset of itself. Furthermore,, as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of is a Fréchet space if and only if so in particular, the strong dual of, which is the space of distributions on, is metrizable.
The three spaces,, and the Schwartz space, as well as the strong duals of each of these three spaces, are complete nuclear Montel bornological spaces, which implies that all six of these locally convex spaces are also paracompact reflexive barrelled Mackey spaces. The spaces and are both distinguished Fréchet spaces. Moreover, both and are Schwartz TVSs.

Convergent sequences

Convergent sequences and their insufficiency to describe topologies
The strong dual spaces of and are sequential spaces but not Fréchet-Urysohn spaces. Moreover, neither the space of test functions nor its strong dual is a sequential space, which in particular implies that their topologies can 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:
  1. There is a compact set containing the supports of all
  2. For each multi-index the sequence of partial derivatives tends uniformly to
Neither the space nor its strong dual is a sequential space, and consequently, their topologies can be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is 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 if and only if it converges pointwise.
Sequences characterize continuity of linear maps valued in locally convex space. Suppose is a locally convex bornological space. Then a linear map into a locally convex space is continuous if and only if it maps null sequences in to bounded subsets of. More generally, such a linear map is continuous if and only if it maps Mackey convergent null sequences 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 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 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 if and only if
for every test function For example, if is the function
and is the distribution corresponding to then
as so 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 ;
  • 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.

Localization of distributions

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. 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. 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 . 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.
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 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:
where the right hand side equals

Extensions and restrictions to an open subset

Let be open subsets of.
Every function can be from its domain to a function on by setting it equal to on the complement This extension is a smooth compactly supported function called the and it will be denoted by.
This assignment defines the operator
which is a continuous injective linear map. It is used to canonically identify as a vector subspace of .
Its transpose
is called the ' 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 trivial extension map is a topological embedding and its range is also dense in its codomain Consequently, if then [|the restriction mapping] is neither injective nor surjective. A distribution is said to be ' if it belongs to the range of the transpose of and it is called if it is extendable to
Unless, the restriction to is neither injective nor surjective.

Spaces of distributions

For all and all, all of the following canonical injections are continuous and have an image/range that is a dense subset of their codomain:
where the topologies on the LB-spaces are the canonical LF topologies as [|defined below].
The range of each of the maps above is dense in the codomain. Indeed, is even sequentially dense in every. For every the canonical inclusion into the normed space is a continuous linear injection and the range of this injection is dense in its codomain if and only if.
Suppose that is one of the LF-spaces or LB-spaces or normed spaces [Lp space|]. Because the canonical injection is a continuous injection whose image is dense in the codomain, this map's transpose is a continuous injection. This injective transpose map thus allows the continuous dual space of to be identified with a certain vector subspace of the space of all distributions. This continuous transpose map is not necessarily a TVS-embedding so the topology that this map transfers from its domain to the image is finer than the subspace topology that this space inherits from.
A linear subspace of carrying a locally convex topology that is finer than the subspace topology induced by is called .
Almost all of the spaces of distributions mentioned in this article arise in this way and any representation theorem about the dual space of may, through the transpose, be transferred directly to elements of the space.

Compactly supported ''Lp''-spaces

Given, the vector space of on and its topology are defined as direct limits of the spaces in a manner analogous to how the canonical LF-topologies on were defined.
For any compact, let denote the set of all element in having a representative whose support is a subset of .
The set is a closed vector subspace and is thus a Banach space and when even a Hilbert space.
Let be the union of all as ranges over all compact subsets of
The set is a vector subspace of whose elements are the compactly supported functions defined on .
Endow with the final topology induced by the inclusion maps as ranges over all compact subsets of.
This topology is called the and it is equal to the final topology induced by any countable set of inclusion maps where are any compact sets with union equal to
This topology makes into an LB-space with a topology that is strictly finer than the norm topology that induces on it.

Radon measures

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection.
Note that the continuous dual space can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals and integral with respect to a Radon measure; that is,
  • if then there exists a Radon measure on such that for all and
  • if is a Radon measure on then the linear functional on defined by is continuous.
Through the injection every Radon measure becomes a distribution on. If is a locally integrable function on then the distribution is a Radon measure; so Radon measures form a large and important space of distributions.
The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally functions in :
Positive Radon measures
A linear function on a space of functions is called if whenever a function that belongs to the domain of is non-negative then. One may show that every positive linear functional on is necessarily continuous.
Lebesgue measure is an example of a positive Radon measure.

Locally integrable functions as distributions

One particularly important class of Radon measures are those that are induced locally integrable functions. The function is called if it is Lebesgue integrable over every compact subset of. This is a large class of functions which includes all continuous functions and all Lp space functions. The topology on is defined in such a fashion that any locally integrable function yields a continuous linear functional on – that is, an element of – denoted here by, whose value on the test function is given by the Lebesgue integral:
Conventionally, one abuses notation by identifying with, provided no confusion can arise, and thus the pairing between and is often written
If and are two locally integrable functions, then the associated distributions and are equal to the same element of if and only if and are equal almost everywhere. In a similar manner, every Radon measure on defines an element of whose value on the test function is As above, it is conventional to abuse notation and write the pairing between a Radon measure and a test function as Conversely, as shown in a theorem by Schwartz, every distribution which is non-negative on non-negative functions is of this form for some Radon measure.
Test functions as distributions
The test functions are themselves locally integrable, and so define distributions. The space of test functions is sequentially dense in with respect to the strong topology on This means that for any there is a sequence of test functions, that converges to when considered as a sequence of distributions. Or equivalently,
Furthermore, is also sequentially dense in the strong dual space of.

Distributions with compact support

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus the image of the transpose, denoted by forms a space of distributions when it is endowed with the strong dual topology of .
The elements of can be identified as the space of distributions with compact support. Explicitly, if is a distribution on then the following are equivalent,
  • ;
  • the support of is compact;
  • the restriction of to when that space is equipped with the subspace topology inherited from, is continuous;
  • there is a compact subset of such that for every test function whose support is completely outside of, we have
Compactly supported distributions define continuous linear functionals on the space ; recall that the topology on is defined such that a sequence of test functions converges to 0 if and only if all derivatives of converge uniformly to 0 on every compact subset of. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from to

Distributions of finite order

Let The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Consequently, the image of denoted by forms a space of distributions when it is endowed with the strong dual topology of . The elements of are ' The distributions of order, which are also called ', are exactly the distributions that are Radon measures.
For a ' is a distribution of order that is not a distribution of order
A distribution is said to be of ' if there is some integer such that it is a distribution of order and the set of distributions of finite order is denoted by Note that if then so that is a vector subspace of and furthermore, if and only if.
Structure of distributions of finite order
Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if is the restriction mapping from to, then the image of under is contained in.
The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:
Example. Let and for every test function let
Then is a distribution of infinite order on. Moreover, can not be extended to a distribution on ; that is, there exists no distribution on such that the restriction of to is equal to.

Tempered distributions and Fourier transform

Defined below are the ', which form a subspace of the space of distributions on This is a proper subspace: while every tempered distribution is a distribution and an element of the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in
Schwartz space
The Schwartz space, is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus is in the Schwartz space provided that any derivative of multiplied with any power of converges to 0 as These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices and define:
Then is in the Schwartz space if all the values satisfy:
The family of seminorms defines a locally convex topology on the Schwartz space. For the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:
Otherwise, one can define a norm on via
The Schwartz space is a Fréchet space. Because the Fourier transform changes into multiplication by and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence in converges to 0 in if and only if the functions converge to 0 uniformly in the whole of which implies that such a sequence must converge to zero in
is dense in The subset of all analytic Schwartz functions is dense in as well.
The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
where represents the completion of the injective tensor product.
Tempered distributions
The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus, the image of the transpose map, denoted by, forms a space of distributions when it is endowed with the strong dual topology of .
The space is called the space of. It is the continuous dual of the Schwartz space. Equivalently, a distribution is a tempered distribution if and only if
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of Lp space for are tempered distributions.
The can also be characterized as, meaning that each derivative of grows at most as fast as some polynomial. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of decays faster than every inverse power of An example of a rapidly falling function is for any positive
Fourier transform
To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform is a TVS-automorphism of the Schwartz space, and the ' is defined to be its transpose which will again be denoted by. So the Fourier transform of the tempered distribution is defined by for every Schwartz function is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
and also with convolution: if is a tempered distribution and is a smooth function on is again a tempered distribution and
is the convolution of and. In particular, the Fourier transform of the constant function equal to 1 is the distribution.
Expressing tempered distributions as sums of derivatives
If is a tempered distribution, then there exists a constant and positive integers and such that for all Schwartz functions
This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function and a multi-index such that
Restriction of distributions to compact sets
If then for any compact set there exists a continuous function compactly supported in and a multi-index such that on.

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:
where and
These definitions associate every and with the continuous linear map:
Moreover, if either has compact support then it also induces a continuous linear map of (resp.
denoted by or is the distribution in defined by:

Schwartz kernel theorem

The tensor product defines a bilinear map
the span of the range of this map is a dense subspace of its codomain. Furthermore, Moreover induces continuous bilinear maps:
where denotes the space of distributions with compact support and is the Schwartz space of rapidly decreasing functions.
This result does not hold for Hilbert spaces such as and its dual space. Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space ? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because is a nuclear space that the Schwartz kernel theorem holds. Like Hilbert spaces, nuclear spaces may be thought as of generalizations of finite dimensional Euclidean space.

Using holomorphic functions as test functions

The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.