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 $f$ is normally thought of as ''acting'' on the ''points'' in its domain by "sending" a point in its domain to the point $f(x).$ Instead of acting on points, distribution theory reinterprets functions such as $f$ 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 $f\; :\; \backslash R\; \backslash to\; \backslash R,$ 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 $f$ "acts on" a test function by "sending" to the number $\backslash textstyle\; \backslash int\_\; fg\; \backslash ,\; dx.$ This new action of $f$ is thus a complex (or real)-valued map, denoted by $D\_,$ whose domain is the space of test functions; this map turns out to have two additional properties$D\_f$ turns out to also be linear and continuous when the space of test functions is given a certain topology called ''the canonical LF topology''. that make it into what is known as a ''distribution on $\backslash R.$'' 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 $U\backslash subset\backslash R^n.$ This space of test functions is denoted by $C\_c^(U)$ or $\backslash mathcal(U)$ and a ''distribution on '' is by definition a linear functional on $C\_c^(U)$ that is continuous when $C\_c^(U)$ is given a topology called ''the canonical LF topology''. This leads to ''the'' space of (all) distributions on , usually denoted by $\backslash mathcal\text{'}(U)$ (note the prime), which by definition is the space of all distributions on $U$ (that is, it is the continuous dual space of $C\_c^(U)$); it is these distributions that are the main focus of this article.
There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If $U\; =\; \backslash R^n$ then the use of Schwartz functionsThe Schwartz space consists of smooth rapidly decreasing test functions, where "rapidly decreasing" means that the function decreases faster than any polynomial increases as points in its domain move away from the origin. as test functions gives rise to a certain subspace of $\backslash mathcal\text{'}(U)$ whose elements are called ''tempered distributions''. These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace of space of distributions $\backslash mathcal\text{'}(U)$ and is thus one example of space of distributions; there are many other spaces of distributions.
There also exist other major classes of test functions that are ''not'' subsets of $C\_c^(U),$ 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.Except for the trivial (i.e. identically ) map, which of course is always analytic. Use of analytic test functions lead to Sato's theory of hyperfunctions.

** History **

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 , generalized functions originated in the work of 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. comments that although the ideas in the transformative book by 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.

Notation

The following notation will be used throughout this article: * $n$ is a fixed positive integer and $U$ is a fixed non-empty open subset of Euclidean space $\backslash R^.$ * $\backslash N\; =\; \backslash $ denotes the natural numbers. * $k$ will denote a non-negative integer or $\backslash infty.$ * If $f$ is a function then $\backslash operatorname(f)$ will denote its domain and the ''support of $f,$'' denoted by $\backslash operatorname(f),$ is defined to be the closure of the set $\backslash $ in $\backslash operatorname(f).$ * For two functions $f,\; g\; :\; U\; \backslash to\; \backslash Complex$, the following notation defines a canonical pairing: ::$\backslash langle\; f,\; g\backslash rangle\; :=\; \backslash int\_U\; f(x)\; g(x)\; \backslash ,dx.$ * A ''multi-index'' of size $n$ is an element in $\backslash N^n$ (given that $n$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be $n$). The ''length'' of a multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$ is defined as $\backslash alpha\_1+\backslash cdots+\backslash alpha\_n$ and denoted by $|\backslash alpha|.$ Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$: ::$\backslash begin\; x^\backslash alpha\; \&=\; x\_1^\; \backslash cdots\; x\_n^\; \backslash \backslash \; \backslash partial^\; \&=\; \backslash frac\; \backslash end$ :We also introduce a partial order of all multi-indices by $\backslash beta\; \backslash ge\; \backslash alpha$ if and only if $\backslash beta\_i\; \backslash ge\; \backslash alpha\_i$ for all $1\; \backslash le\; i\backslash le\; n.$ When $\backslash beta\; \backslash ge\; \backslash alpha$ we define their multi-index binomial coefficient as: ::$\backslash binom\; :=\; \backslash binom\; \backslash cdots\; \backslash binom.$ * $\backslash mathbb$ will denote a certain non-empty collection of compact subsets of $U$ (described in detail below).

** 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 $\backslash R^n$ with any (paracompact) smooth manifold.
Note that for all $j,\; k\; \backslash in\; \backslash $ and any compact subsets and of , we have:
:$\backslash begin\; C^k(K)\; \&\backslash subseteq\; C^k\_c(U)\; \backslash subseteq\; C^k(U)\; \backslash \backslash \; C^k(K)\; \&\backslash subseteq\; C^k(L)\; \&\&\; \backslash text\; K\; \backslash subseteq\; L\; \backslash \backslash \; C^k(K)\; \&\backslash subseteq\; C^j(K)\; \&\&\; \backslash text\; j\; \backslash le\; k\; \backslash \backslash \; C\_c^k(U)\; \&\backslash subseteq\; C^j\_c(U)\; \&\&\; \backslash text\; j\; \backslash le\; k\; \backslash \backslash \; C^k(U)\; \&\backslash subseteq\; C^j(U)\; \&\&\; \backslash text\; j\; \backslash le\; k\; \backslash \backslash \; \backslash end$
Distributions on are defined to be the continuous linear functionals on $C\_c^(U)$ 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 is a linear functional on $C\_c^(U)$ then the is a distribution if and only if the following equivalent conditions are satisfied:
# For every compact subset $K\backslash subseteq\; U$ there exist constants $C>0$ and $N\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C^(K),$
#:$|\; T(f)|\; \backslash le\; C\; \backslash sup\; \backslash ;$
# For every compact subset $K\backslash subseteq\; U$ there exist constants $C\_K>0$ and $N\_K\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C\_c^(U)$ with support contained in $K,$
#:$|T(f)|\; \backslash le\; C\_K\; \backslash sup\; \backslash ;$
# For any compact subset $K\backslash subseteq\; U$ and any sequence $\backslash \_^\backslash infty$ in $C^(K),$ if $\backslash \_^$ converges uniformly to zero on $K$ for all multi-indices $\backslash alpha$, then $T(f\_i)\; \backslash to\; 0.$
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 $C\_c^(U)$ and $\backslash mathcal(U).$
To define the space of distributions we must first define the canonical LF-topology, which in turn requires that several other topological vector spaces (TVSs) be defined first. We will first define a topology on $C^(U),$ then assign every $C^(K)$ the subspace topology induced on it by $C^(U),$ and finally we define the canonical LF-topology on $C\_c^(U).$ We use the canonical LF-topology to define a topology on the space of distributions, which permits us to consider things such as convergence of distributions.
;Choice of compact sets
Throughout, will be any collection of compact subsets of such that (1) $U\; =\; \backslash cup\_\; K,$ and (2) for any compact there exists some such that . The most common choices for are:
* The set of all compact subsets of , or
* A set $\backslash left\backslash $ where $U\; =\; \backslash cup\_^\; U\_i,$ and for all , $\backslash overline\; \backslash subseteq\; U\_$ and is a relatively compact non-empty open subset of (i.e. "relatively compact" means that the closure of , in either or $\backslash R^n,$ is compact).
We make into a directed set by defining if and only if . Note that although the definitions of the subsequently defined topologies explicitly reference , in reality they do not depend on the choice of ; that is, if and are any two such collections of compact subsets of , then the topologies defined on $C^k(U)$ and $C\_c^k(U)$ 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 $C^k(U).$ 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.
Each of the functions above are non-negative -valuedThe image of the compact set under a continuous -valued map (e.g. $x\; \backslash mapsto\; \backslash left|\; \backslash partial^p\; f(x)\; \backslash $ for ) is itself a compact, and thus bounded, subset of . If then this implies that each of the functions defined above is -valued (i.e. none of the supremums above are ever equal to ). seminorms on $C^k(U).$
Each of the following families of seminorms generates the same locally convex vector topology on $C^k(U)$:
:$\backslash begin\; (1)\; \backslash quad\; \&\backslash \; \backslash \backslash \; (2)\; \backslash quad\; \&\backslash \; \backslash \backslash \; (3)\; \backslash quad\; \&\backslash \; \backslash \backslash \; (4)\; \backslash quad\; \&\backslash \; \backslash end$
With this topology, $C^k(U)$ 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 $C^k(U).$
Under this topology, a net $(f\_i)\_$ in $C^k(U)$ converges to $f\; \backslash in\; C^k(U)$ if and only if for every multi-index with and every , the net $(\backslash partial^p\; f\_i)\_$ converges to $\backslash partial^p\; f$ uniformly on . For any $k\; \backslash in\; \backslash ,$ any bounded subset of $C^(U)$ is a relatively compact subset of $C^k(U).$ In particular, a subset of $C^(U)$ is bounded if and only if it is bounded in $C^i(U)$ for all $i\backslash in\; \backslash N.$ The space $C^k(U)$ is a Montel space if and only if .
The topology on $C^(U)$ is the superior limit of the subspace topologies induced on $C^(U)$ by the TVSs $C^i(U)$ as ranges over the non-negative integers. A subset of $C^(U)$ is open in this topology if and only if there exists $i\backslash in\; \backslash N$ such that is open when $C^(U)$ is endowed with the subspace topology induced by $C^i(U).$
;Metric defining the topology
If the family of compact sets $\backslash mathbb\; =\; \backslash left\backslash $ satisfies $U\; =\; \backslash cup\_^\; U\_i$ and $\backslash overline\; \backslash subseteq\; U\_$ for all , then a complete translation-invariant metric on $C^(U)$ can be obtained by taking a suitable countable Fréchet combination of any one of the above families.
For example, using the seminorms $\backslash left(\; r\_\; \backslash right)\_^$ results in
:$d(\; f,\; g\; )\; :=\; \backslash sum^\_\; \backslash frac\; \backslash frac\; =\; \backslash sum^\_\; \backslash frac\; \backslash frac.$
Often, it is easier to just consider seminorms.

** Topology on C**^{k}(K)

As before, fix $k\; \backslash in\; \backslash .$ Recall that if $K$ is any compact subset of $U$ then $C^k(K)\; \backslash subseteq\; C^k(U).$
For any compact subset , $C^k(K)$ is a closed subspace of the Fréchet space $C^k(U)$ and is thus also a Fréchet space. For all compact with , denote the natural inclusion by $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C^k(L).$ Then this map is a linear embedding of TVSs (i.e. a linear map that is also a topological embedding) whose range is closed in its codomain; said differently, the topology on $C^k(K)$ is identical to the subspace topology it inherits from $C^k(L),$ and also $C^k(K)$ is a closed subset of $C^k(L).$ The interior of $C^(K)$ relative to $C^(U)$ is empty.
If $k$ is finite then $C^k(K)$ is a Banach space with a topology that can be defined by the norm
:$r\_(f):=\backslash sup\_\; \backslash left\; (\; \backslash sup\_\; \backslash left|\; \backslash partial^p\; f(x\_0)\; \backslash \; \backslash right).$
And when , then $C^k(K)$ is even a Hilbert space. The space $C^(K)$ is a distinguished Schwartz Montel space so if $C^(K)\backslash neq\backslash $ then it is ''not'' normable and thus ''not'' a Banach space (although like all other $C^k(K),$ it is a Fréchet space).

** Trivial extensions and independence of ''C''**^{''k''}(''K'')'s topology from ''U''

The definition of $C^k(K)$ depends on so we will let $C^k(K;U)$ denote the topological space $C^k(K),$ which by definition is a topological subspace of $C^k(U).$ Suppose is an open subset of $\backslash R^n$ containing $U.$ Given $f\; \backslash in\; C\_c^k(U),$ its is by definition, the function $F\; :\; V\; \backslash to\; \backslash Complex$ defined by:
:$F(x)\; =\; \backslash begin\; f(x)\; \&\; x\; \backslash in\; U,\; \backslash \backslash \; 0\; \&\; \backslash text,\; \backslash end$
so that $F\; \backslash in\; C^k(V).$ Let $I\; :\; C\_c^k(U)\; \backslash to\; C^k(V)$ denote the map that sends a function in $C\_c^k(U)$ to its trivial extension on . This map is a linear injection and for every compact subset $K\; \backslash subseteq\; U,$ we have $I\backslash left(\; C^k(K;\; U)\; \backslash right)\; =\; C^k(K;\; V),$ where $C^k(K;\; V)$ is the vector subspace of $C^k(V)$ consisting of maps with support contained in (since , is a compact subset of as well). It follows that $I\backslash left(\; C\_c^k(U)\; \backslash right)\; \backslash subseteq\; C\_c^k(V).$ If is restricted to $C^k(K;\; U)$ then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism):
:$C^k(K;\; U)\; \backslash to\; C^k(K;V)$
and thus the next two maps (which like the previous map are defined by $f\; \backslash mapsto\; I(f)$) are topological embeddings:
:$C^k(K;\; U)\; \backslash to\; C^k(V),$
:$C^k(K;\; U)\; \backslash to\; C\_c^k(V),$
(the topology on $C\_c^k(V)$ is the canonical LF topology, which is defined later). Using $C\_c^k(U)\; \backslash ni\; f\; \backslash mapsto\; I(f)\; \backslash in\; C\_c^k(V)$ we identify $C\_c^k(U)$ with its image in $C\_c^k(V)\; \backslash subseteq\; C^k(V).$ Because $C^k(K;\; U)\; \backslash subseteq\; C\_c^k(U),$ through this identification, $C^k(K;\; U)$ can also be considered as a subset of $C^k(V).$ Importantly, the subspace topology $C^k(K;\; U)$ inherits from $C^k(U)$ (when it is viewed as a subset of $C^k(U)$) is identical to the subspace topology that it inherits from $C^k(V)$ (when $C^k(K;\; U)$ is viewed instead as a subset of $C^k(V)$ via the identification). Thus the topology on $C^k(K;U)$ is independent of the open subset of $\backslash R^n$ that contains . This justifies the practice of written $C^k(K)$ instead of $C^k(K;\; U).$

** Topology on the spaces of test functions and distributions **

Recall that $C\_c^k(U)$ denote all those functions in $C^k(U)$ that have compact support in , where note that $C\_c^k(U)$ is the union of all $C^k(K)$ as ranges over . Moreover, for every , $C\_c^k(U)$ is a dense subset of $C^k(U).$ The special case when gives us the space of test functions.

** Canonical LF topology **

We now define 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.
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 (we say that such a collection is ''directed by subset inclusion''). For all compact with , there are natural inclusions
:$\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C^k(L)\backslash quad\; \backslash text\; \backslash quad\; \backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C\_c^k(U).$
Recall from above that the map $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C^k(L)$ is a topological embedding. The collection of maps
:$\backslash left\; \backslash $
forms a direct system in the category of locally convex topological vector spaces that is directed by (under subset inclusion). This system's direct limit (in the category of locally convex TVSs) is the pair $(C\_c^k(U),\; \backslash operatorname\_^)$ where $\backslash operatorname\_^\; :=\; \backslash left(\backslash operatorname\_^\backslash right)\_$ are the natural inclusions and where $C\_c^k(U)$ is now endowed with the (unique) strongest locally convex topology making all of the inclusion maps $\backslash operatorname\_^\; =\; \backslash left(\backslash operatorname\_^\backslash right)\_$ continuous.
;Neighborhoods of the origin
If is a convex subset of $C\_c^k(U),$ 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 $C\_c^k(U).$ Since the topology of any topological vector space is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually ''define'' the canonical LF topology by declaring that a convex balanced subset is a neighborhood of the origin if and only if it satisfies condition .
;Topology defined via differential operators
A ''linear differential operator in with smooth coefficients'' is a sum
:$P\; :=\; \backslash sum\_\; c\_\; \backslash partial^$
where $c\_\; \backslash in\; C^(U)$ and all but finitely many of $c\_$ are identically . The integer $\backslash sup\; \backslash $ is called the ''order'' of the differential operator $P.$ If $P$ is a linear differential operator of order then it induces a canonical linear map $C^k(U)\; \backslash to\; C^0(U)$ defined by $\backslash phi\; \backslash mapsto\; P\backslash phi,$ where we shall reuse notation and also denote this map by $P.$
For any , the canonical LF topology on $C\_c^k(U)$ is the weakest locally convex TVS topology making all linear differential operators in of order into continuous maps from $C\_c^k(U)$ into $C\_c^0(U).$

= Basic properties

= ;Canonical LF topology's independence from One benefit of defining the canonical LF topology as the direct limit of a direct system is that we may immediately use the universal property of direct limits. Another benefit is that we can use well-known results from category theory to deduce that the canonical LF topology is actually independent of the particular choice of the directed collection of compact sets. And by considering different collections (in particular, those mentioned at the beginning of this article), we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes $C\_c^k(U)$ into a Hausdorff locally convex strict LF-space (and also a strict LB-space if ), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).If we take to be the set of ''all'' compact subsets of then we can use the universal property of direct limits to conclude that the inclusion $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C\_c^k(U)$ is a continuous and even that they are topological embedding for every compact subset . If however, we take to be the set of closures of some countable increasing sequence of relatively compact open subsets of having all of the properties mentioned earlier in this in this article then we immediately deduce that $C\_c^k(U)$ is a Hausdorff locally convex strict LF-space (and even a strict LB-space when ). All of these facts can also be proved directly without using direct systems (although with more work). ;Universal property From the universal property of direct limits, we know that if $u\; :\; C\_c^k(U)\; \backslash to\; Y$ is a linear map into a locally convex space (not necessarily Hausdorff), then is continuous if and only if is bounded if and only if for every , the restriction of to $C^k(K)$ is continuous (or bounded). ;Dependence of the canonical LF topology on Suppose is an open subset of $\backslash R^n$ containing $U.$ Let $I:\; C\_c^k(U)\backslash to\; C\_c^k(V)$ denote the map that sends a function in $C\_c^k(U)$ to its trivial extension on (which was defined above). This map is a continuous linear map. If (and only if) then $I(C\_c^(U))$ is ''not'' a dense subset of $C\_c^(V)$ and $I:\; C\_c^(U)\backslash to\; C\_c^(V)$ is ''not'' a topological embedding. Consequently, if then the transpose of $I:\; C\_c^(U)\backslash to\; C\_c^(V)$ is neither one-to-one nor onto. ;Bounded subsets A subset of $C\_c^k(U)$ is bounded in $C\_c^k(U)$ if and only if there exists some such that $B\; \backslash subseteq\; C^k(K)$ and is a bounded subset of $C^k(K).$ Moreover, if is compact and $S\; \backslash subseteq\; C^k(K)$ then is bounded in $C^k(K)$ if and only if it is bounded in $C^k(U).$ For any , any bounded subset of $C\_c^(U)$ (resp. $C^(U)$) is a relatively compact subset of $C\_c^k(U)$ (resp. $C^k(U)$), where . ;Non-metrizability For all compact , the interior of $C^k(K)$ in $C\_c^k(U)$ is empty so that $C\_c^k(U)$ is of the first category in itself. It follows from Baire's theorem that $C\_c^k(U)$ is ''not'' metrizable and thus also ''not'' normable (see this footnoteFor any TVS (metrizable or otherwise), the notion of completeness depends entirely on a certain so-called "canonical uniformity" that is defined using ''only'' the subtraction operation (see the article Complete topological vector space for more details). In this way, the notion of a complete TVS does not ''require'' the existence of any metric. However, if the TVS is metrizable and if is ''any'' translation-invariant metric on that defines its topology, then is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if the is a complete metric space. So if a TVS happens to have a topology that can be defined by such a metric then may be used to deduce the completeness of but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the Cartesian product of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS $\backslash R^\backslash N,$ which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on $\backslash R^\backslash N$). for an explanation of how the non-metrizable space $C\_c^k(U)$ can be complete even thought it does not admit a metric). The fact that $C\_c^(U)$ is a nuclear Montel space makes up for the non-metrizability of $C\_c^(U)$ (see this footnote for a more detailed explanation).One reason for giving $C\_c^(U)$ the canonical LF topology is because it is with this topology that $C\_c^(U)$ and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of topological vector spaces (TVSs) that are particularly similar to finite-dimensional Euclidean spaces: the Banach spaces (especially Hilbert spaces) and the nuclear Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are nuclear Montel Hilbert spaces but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. ''infinite'' differentiability, such as $C\_c^(U)$ and $C^(U)$) end up being nuclear TVSs while TVSs associated with ''finite'' continuous differentiability (such as $C^k(K)$ with compact and ) often end up being non-nuclear spaces, such as Banach spaces. ;Relationships between spaces Using the universal property of direct limits and the fact that the natural inclusions $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C^k(L)$ are all topological embedding, one may show that all of the maps $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C\_c^k(U)$ are also topological embeddings. Said differently, the topology on $C^k(K)$ is identical to the subspace topology that it inherits from $C\_c^k(U),$ where recall that $C^k(K)$'s topology was ''defined'' to be the subspace topology induced on it by $C^k(U).$ In particular, both $C\_c^k(U)$ and $C^k(U)$ induces the same subspace topology on $C^k(K).$ However, this does ''not'' imply that the canonical LF topology on $C\_c^k(U)$ is equal to the subspace topology induced on $C\_c^k(U)$ by $C^k(U)$; these two topologies on $C\_c^k(U)$ are in fact ''never'' equal to each other since the canonical LF topology is ''never'' metrizable while the subspace topology induced on it by $C^k(U)$ is metrizable (since recall that $C^k(U)$ is metrizable). The canonical LF topology on $C\_c^k(U)$ is actually ''strictly finer'' than the subspace topology that it inherits from $C^k(U)$ (thus the natural inclusion $C\_c^k(U)\backslash to\; C^k(U)$ is continuous but ''not'' a topological embedding). Indeed, the canonical LF topology is so fine that if $C\_c^(U)\backslash to\; X$ denotes some linear map that is a "natural inclusion" (such as $C\_c^(U)\backslash to\; C^k(U),$ or $C\_c^(U)\backslash to\; L^p(U),$ or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions, Radon measures, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on $C\_c^(U),$ the fine nature of the canonical LF topology means that more linear functionals on $C\_c^(U)$ end up being continuous ("more" means as compared to a coarser topology that we could have placed on $C\_c^(U)$ such as for instance, the subspace topology induced by some $C^k(U),$ which although it would have made $C\_c^(U)$ metrizable, it would have also resulted in fewer linear functionals on $C\_c^(U)$ being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making $C\_c^(U)$ into a complete TVS). ;Other properties * The differentiation map $C\_c^(U)\; \backslash to\; C\_c^(U)$ is a surjective continuous linear operator. * The bilinear multiplication map $C^(\backslash R^m)\; \backslash times\; C\_c^(\backslash R^n)\; \backslash to\; C\_c^(\backslash R^)$ given by $(f,g)\backslash mapsto\; fg$ is ''not'' continuous; it is however, hypocontinuous.

** Distributions **

As discussed earlier, continuous linear functionals on a $C\_c^(U)$ are known as distributions on . Thus the set of all distributions on is the continuous dual space of $C\_c^(U),$ which when endowed with the strong dual topology is denoted by $\backslash mathcal\text{'}(U).$
We have the canonical duality pairing between a distribution on and a test function $f\; \backslash in\; C\_c^(U),$ which is denoted using angle brackets by
:$\backslash begin\; \backslash mathcal\text{'}(U)\; \backslash times\; C\_c^(U)\; \backslash to\; \backslash R\; \backslash \backslash \; (T,\; f)\; \backslash mapsto\; \backslash langle\; T,\; f\; \backslash rangle\; :=\; T(f)\; \backslash end$
One interprets this notation as the distribution acting on the test function $f$ to give a scalar, or symmetrically as the test function $f$ acting on the distribution .
;Characterizations of distributions
Proposition. If is a linear functional on $C\_c^(U)$ then the following are equivalent:
# is a distribution;
# (''definition'') is continuous;
# is continuous at the origin;
# is uniformly continuous;
# is a bounded operator;
# is sequentially continuous;
#* explicitly, for every sequence $\backslash left(\; f\_i\; \backslash right)\_^\backslash infty$ in $C\_c^(U)$ that converges in $C\_c^(U)$ to some $f\; \backslash in\; C\_c^(U),$ $\backslash lim\_\; T\backslash left(\; f\_i\; \backslash right)\; =\; T(f);$Even though the topology of $C\_c^(U)$ is not metrizable, a linear functional on $C\_c^(U)$ is continuous if and only if it is sequentially continuous.
# is sequentially continuous at the origin; in other words, maps null sequences to null sequences;
#* explicitly, for every sequence $\backslash left(\; f\_i\; \backslash right)\_^\backslash infty$ in $C\_c^(U)$ that converges in $C\_c^(U)$ to the origin (such a sequence is called a ''null sequence''), $\backslash lim\_\; T\backslash left(\; f\_i\; \backslash right)\; =\; 0;$
#* a ''null sequence'' is by definition a sequence that converges to the origin;
# maps null sequences to bounded subsets;
#* explicitly, for every sequence $\backslash left(\; f\_i\; \backslash right)\_^\backslash infty$ in $C\_c^(U)$ that converges in $C\_c^(U)$ to the origin, the sequence $\backslash left(\; T\backslash left(\; f\_i\; \backslash right)\; \backslash right)\_^$ is bounded;
# maps Mackey convergence null sequences to bounded subsets;
#* explicitly, for every Mackey convergent null sequence $\backslash left(\; f\_i\; \backslash right)\_^\backslash infty$ in $C\_c^(U),$ the sequence $\backslash left(\; T\backslash left(\; f\_i\; \backslash right)\; \backslash right)\_^$ is bounded;
#* a sequence is said to be ''Mackey convergent to '' 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 (in the usual sense);
# The kernel of is a closed subspace of $C\_c^(U);$
# The graph of is a closed;
# There exists a continuous seminorm on $C\_c^(U)$ such that $|T|\backslash le\; g;$
# There exists a constant , a collection of continuous seminorms, $\backslash mathcal,$ that defines the canonical LF topology of $C\_c^(U),$ and a finite subset $\backslash \; \backslash subseteq\; \backslash mathcal$ such that $|T|\; \backslash le\; C(g\_1\; +\; \backslash cdots\; g\_m);$If $\backslash mathcal$ is also directed under the usual function comparison then we can take the finite collection to consist of a single element.
# For every compact subset $K\backslash subseteq\; U$ there exist constants $C>0$ and $N\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C^(K),$
#:$|\; T(f)|\; \backslash le\; C\; \backslash sup\; \backslash ;$
# For every compact subset $K\backslash subseteq\; U$ there exist constants $C\_K>0$ and $N\_K\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C\_c^(U)$ with support contained in $K,$
#:$|T(f)|\; \backslash le\; C\_K\; \backslash sup\; \backslash ;$
# For any compact subset $K\backslash subseteq\; U$ and any sequence $\backslash \_^\backslash infty$ in $C^(K),$ if $\backslash \_^$ converges uniformly to zero for all multi-indices , then $T(f\_i)\; \backslash to\; 0;$
# Any of the ''three'' statements immediately above (i.e. statements 14, 15, and 16) 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 ''the strong dual topology''.In functional analysis, the strong dual topology is often the "standard" or "default" topology placed on the continuous dual space $X\text{'},$ where if is a normed space then this strong dual topology is the same as the usual norm-induced topology on $X\text{'}.$ This topology is chosen because it is with this topology that $\backslash mathcal\text{'}(U)$ becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds. No matter what dual topology is placed on $\backslash mathcal\text{'}(U)$,Technically, the topology must be coarser than the strong dual topology and also simultaneously be finer that the weak* topology. a ''sequence'' of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net). No matter which topology is chosen, $\backslash mathcal\text{'}(U)$ will be a non-metrizable, locally convex topological vector space. The space $\backslash mathcal\text{'}(U)$ is separable and has the strong Pytkeev propertyGabriyelyan, S.S. Kakol J., and·Leiderman, A

"The strong Pitkeev property for topological groups and topological vector spaces"

/ref> 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 ''not'' be defined using only sequences.

** Topological properties **

;Topological vector space categories
The canonical LF topology makes $C\_c^k(U)$ into a complete distinguished strict LF-space (and a strict LB-space if and only if ), which implies that $C\_c^k(U)$ is a meager subset of itself. Furthermore, $C\_c^k(U),$ as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of $C\_c^k(U)$ is a Fréchet space if and only if so in particular, the strong dual of $C\_c^(U),$ which is the space $\backslash mathcal\text{'}(U)$ of distributions on , is ''not'' metrizable (note that the weak-* topology on $\backslash mathcal\text{'}(U)$ also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives $\backslash mathcal\text{'}(U)$).
The three spaces $C\_c^(U),$ $C^(U),$ and the Schwartz space $\backslash mathcal(\backslash R^n),$ 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 $C^(U)$ and $\backslash mathcal(\backslash R^n)$ are both distinguished Fréchet spaces. Moreover, both $C\_c^(U)$ and $\backslash mathcal(\backslash R^n)$ are Schwartz TVSs.

** Convergent sequences **

;Convergent sequences and their insufficiency to describe topologies
The strong dual spaces of $C^(U)$ and $\backslash mathcal(\backslash R^n)$ are sequential spaces but not Fréchet-Urysohn spaces. Moreover, neither the space of test functions $C\_c^(U)$ nor its strong dual $\backslash mathcal\text{'}(U)$ is a sequential space (not even an Ascoli space),Gabriyelyan, Saa

"Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"

(2017)T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36. which in particular implies that their topologies can ''not'' be defined entirely in terms of convergent sequences. A sequence $(f\_i)\_^$ in $C\_c^k(U)$ converges in $C\_c^k(U)$ if and only if there exists some such that $C^k(K)$ contains this sequence and this sequence converges in $C^k(K)$; equivalently, it converges if and only if the following two conditions hold: # There is a compact set containing the supports of all $f\_i.$ # For each multi-index , the sequence of partial derivatives $\backslash partial^\backslash alpha\; f\_$ tends uniformly to $\backslash partial^\backslash alpha\; f.$ Neither the space $C\_c^(U)$ nor its strong dual $\backslash mathcal\text{'}(U)$ is a sequential space, and consequently, their topologies can ''not'' be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is ''not'' enough to define the canonical LF topology on $C\_c^(U).$ The same can be said of the strong dual topology on $\backslash mathcal\text{'}(U).$ ;What sequences do characterize Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology, which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually ''define'' the convergence of a sequence of distributions; this is fine for sequences but it does ''not'' extend to the convergence of nets of distributions since a net may converge pointwise but fail to convergence in the strong dual topology). Sequences characterize continuity of linear maps valued in locally convex space. Suppose is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map into a locally convex space is continuous if and only if it maps null sequencesA ''null sequence'' is a sequence that converges to the origin. in to bounded subsets of .Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences. More generally, such a linear map is continuous if and only if it maps Mackey convergent null sequencesA sequence is said to be ''Mackey convergent to in $X,$'' if there exists a divergent sequence of positive real number such that is a bounded set in $X.$ to bounded subsets of $Y.$ So in particular, if a linear map into a locally convex space is sequentially continuous at the origin then it is continuous. However, this does ''not'' necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs. For every $k\; \backslash in\; \backslash ,\; C\_c^(U)$ is sequentially dense in $C\_c^k(U).$ Furthermore, $\backslash $ is a sequentially dense subset of $\backslash mathcal\text{'}(U)$ (with its strong dual topology) and also a sequentially dense subset of the strong dual space of $C^(U).$ ;Sequences of distributions A sequence of distributions $(T\_i)\_^$ converges with respect to the weak-* topology on $\backslash mathcal\text{'}(U)$ to a distribution if and only if :$\backslash langle\; T\_,\; f\; \backslash rangle\; \backslash to\; \backslash langle\; T,\; f\; \backslash rangle$ for every test function $f\; \backslash in\; \backslash mathcal(U).$ For example, if $f\_m:\backslash R\backslash to\backslash R$ is the function :$f\_m(x)\; =\; \backslash begin\; m\; \&\; \backslash text\; x\; \backslash in,\backslash frac\backslash \backslash \; 0\; \&\; \backslash text\; \backslash end$ and is the distribution corresponding to $f\_m,$ then :$\backslash langle\; T\_m,\; f\; \backslash rangle\; =\; m\; \backslash int\_0^\; f(x)\backslash ,\; dx\; \backslash to\; f(0)\; =\; \backslash langle\; \backslash delta,\; f\; \backslash rangle$ as , so in $\backslash mathcal\text{'}(\backslash R).$ Thus, for large , the function $f\_m$ can be regarded as an approximation of the Dirac delta distribution. ;Other properties * The strong dual space of $\backslash mathcal\text{'}(U)$ is TVS isomorphic to $C\_c^(U)$ via the canonical TVS-isomorphism $C\_c^(U)\; \backslash to\; (\backslash mathcal\text{'}(U))\text{'}\_$ defined by sending $f\; \backslash in\; C\_c^(U)$ to ''value at $f$'' (i.e. to the linear functional on $\backslash mathcal\text{'}(U)$ defined by sending $d\; \backslash in\; \backslash mathcal\text{'}(U)$ to $d(f)$); * On any bounded subset of $\backslash mathcal\text{'}(U),$ the weak and strong subspace topologies coincide; the same is true for $C\_c^(U)$; * Every weakly convergent sequence in $\backslash mathcal\text{'}(U)$ is strongly convergent (although this does not extend to nets).

** Localization of distributions **

There is no way to define the value of a distribution in $\backslash mathcal\text{'}(U)$ at a particular point of . However, as is the case with functions, distributions on restrict to give distributions on open subsets of . Furthermore, distributions are ''locally determined'' in the sense that a distribution on all of can be assembled from a distribution on an open cover of satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.

** Restrictions to an open subset **

Let and be open subsets of $\backslash R^n$ with . Let $E\_:\; \backslash mathcal(V)\; \backslash to\; \backslash mathcal(U)$ be the operator which ''extends by zero'' a given smooth function compactly supported in to a smooth function compactly supported in the larger set . The transpose of $E\_$ is called the restriction mapping and is denoted by $\backslash rho\_\; :=\; ^E\_\; :\; \backslash mathcal\text{'}(U)\; \backslash to\; \backslash mathcal\text{'}(V).$
The map $E\_\; :\; \backslash mathcal(V)\; \backslash to\; \backslash mathcal(U)$ is a continuous injection where if then it is ''not'' a topological embedding and its range is ''not'' dense in $\backslash mathcal(U),$ which implies that this map's transpose is neither injective nor surjective and that the topology that $E\_$ transfers from $\backslash mathcal(V)$ onto its image is strictly finer than the subspace topology that $\backslash mathcal(U)$ induces on this same set. A distribution $S\; \backslash in\; \backslash mathcal\text{'}(V)$ is said to be ''extendible to '' if it belongs to the range of the transpose of $E\_$ and it is called ''extendible'' if it is extendable to $\backslash R^n.$
For any distribution $T\; \backslash in\; \backslash mathcal\text{'}(U),$ the restriction is a distribution in $\backslash mathcal\text{'}(V)$ defined by:
:$\backslash qquad\; \backslash langle\; \backslash rho\_\; T,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T,\; E\_\; \backslash phi\; \backslash rangle\; \backslash quad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(V).$
Unless , the restriction to is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of . For instance, if and , then the distribution
:$T(x)\; =\; \backslash sum\_^\; n\; \backslash ,\; \backslash delta\backslash left(x-\backslash frac\backslash right)$
is in $\backslash mathcal\text{'}(V)$ but admits no extension to $\backslash mathcal\text{'}(U).$

** Gluing and distributions that vanish in a set **

Let be an open subset of . $T\; \backslash in\; \backslash mathcal\text{'}(U)$ is said to ''vanish in '' if for all $f\; \backslash in\; \backslash mathcal(U)$ such that $\backslash operatorname(f)\; \backslash subseteq\; V$ we have $Tf\; =\; 0.$ vanishes in if and only if the restriction of to is equal to 0, or equivalently, if and only if lies in the kernel of the restriction map .
:Corollary. Let $(U\_i)\_$ be a collection of open subsets of $\backslash R^n$ and let $T\; \backslash in\; \backslash mathcal\text{'}(\backslash cup\_\; U\_i).$ if and only if for each $i\; \backslash in\; I,$ the restriction of to $U\_i$ is equal to 0.
:Corollary. The union of all open subsets of in which a distribution vanishes is an open subset of in which vanishes.

** Support of a distribution **

This last corollary implies that for every distribution on , there exists a unique largest subset of such that vanishes in (and does not vanish in any open subset of that is not contained in ); the complement in of this unique largest open subset is called ''the support of ''. Thus
:$\backslash operatorname(T)\; =\; U\; \backslash setminus\; \backslash bigcup\; \backslash .$
If $f$ is a locally integrable function on and if $D\_f$ is its associated distribution, then the support of $D\_f$ is the smallest closed subset of in the complement of which $f$ is almost everywhere equal to 0. If $f$ is continuous, then the support of $D\_f$ is equal to the closure of the set of points in at which $f$ does not vanish. The support of the distribution associated with the Dirac measure at a point $x\_0$ is the set $\backslash .$ If the support of a test function $f$ does not intersect the support of a distribution then . A distribution is 0 if and only if its support is empty. If $f\; \backslash in\; C^(U)$ is identically 1 on some open set containing the support of a distribution then . If the support of a distribution is compact then it has finite order and furthermore, there is a constant ''C'' and a non-negative integer ''N'' such that:
:$\backslash qquad\; |T\; \backslash phi|\; \backslash leq\; C\backslash |\backslash phi\backslash |\_\; :=\; C\; \backslash sup\; \backslash left\backslash \; \backslash quad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(U).$
If has compact support then it has a unique extension to a continuous linear functional $\backslash widehat$ on $C^(U)$; this functional can be defined by $\backslash widehat\; (f)\; :=\; T(\backslash psi\; f),$ where $\backslash psi\; \backslash in\; \backslash mathcal(U)$ is any function that is identically 1 on an open set containing the support of .
If $S,\; T\; \backslash in\; \backslash mathcal\text{'}(U)$ and $\backslash lambda\; \backslash neq\; 0$ then $\backslash operatorname(S\; +\; T)\; \backslash subseteq\; \backslash operatorname(S)\; \backslash cup\; \backslash operatorname(T)$ and $\backslash operatorname(\backslash lambda\; T)\; =\; \backslash operatorname(T).$ Thus, distributions with support in a given subset $A\; \backslash subseteq\; U$ form a vector subspace of $\backslash mathcal\text{'}(U)$; such a subspace is weakly closed in $\backslash mathcal\text{'}(U)$ if and only if ''A'' is closed in . Furthermore, if $P$ is a differential operator in , then for all distributions on and all $f\; \backslash in\; C^(U)$ we have $\backslash operatorname\; (P(x,\; \backslash partial)T)\; \backslash subseteq\; \backslash operatorname(T)$ and $\backslash operatorname(fT)\; \backslash subseteq\; \backslash operatorname(f)\; \backslash cap\; \backslash operatorname(T).$

** Distributions with compact support **

;Support in a point set and Dirac measures
For any $x\; \backslash in\; U,$ let $\backslash delta\_x\; \backslash in\; \backslash mathcal\text{'}(U)$ denote the distribution induced by the Dirac measure at ''x''. For any $x\_0\; \backslash in\; U$ and distribution $T\; \backslash in\; \backslash mathcal\text{'}(U),$ the support of is contained in $\backslash $ if and only if is a finite linear combination of derivatives of the Dirac measure at $x\_0.$ If in addition the order of is $\backslash leq\; k$ then there exist constants $\backslash alpha\_p$ such that:
:$T\; =\; \backslash sum\_\; \backslash alpha\_p\; \backslash partial^\; \backslash delta\_.$
Said differently, if has support at a single point $\backslash ,$ then is in fact a finite linear combination of distributional derivatives of the function at . That is, there exists an integer and complex constants such that
:$T\; =\; \backslash sum\_\; a\_\backslash alpha\; \backslash partial^\backslash alpha(\backslash tau\_P\backslash delta)$
where $\backslash tau\_P$ is the translation operator.
;Distribution with compact support
;Distributions of finite order with support in an open subset

** Global structure of distributions **

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of $\backslash mathcal(U)$ (or the Schwartz space $\backslash mathcal(\backslash R^n)$ 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

** Decomposition of distributions as sums of derivatives of continuous functions **

By combining the above results, one may express any distribution on 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 . In other words for arbitrary $T\; \backslash in\; \backslash mathcal\text{'}(U)$ we can write:
:$T\; =\; \backslash sum\_^\; \backslash sum\_\; \backslash partial^\; f\_,$
where $P\_1,\; P\_2,\; \backslash ldots$ are finite sets of multi-indices and the functions $f\_$ are continuous.
Note that the infinite sum above is well-defined as a distribution. The value of for a given $f\; \backslash in\; \backslash mathcal(U)$ can be computed using the finitely many that intersect the support of $f.$

** Operations on distributions **

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if $A:\backslash mathcal(U)\backslash to\backslash mathcal(U)$ is a linear map which is continuous with respect to the weak topology, then it is possible to extend to a map $A\; :\; \backslash mathcal\text{'}(U)\backslash to\; \backslash mathcal\text{'}(U)$ by passing to the limit.This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.

** 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 $A:\; X\; \backslash to\; Y$ is the linear map $^A\; :\; Y\text{'}\; \backslash to\; X\text{'}$ defined by $^A(y\text{'})\; :=\; y\text{'}\; \backslash circ\; A,$ or equivalently, it is the unique map satisfying $\backslash langle\; y\text{'},\; A(x)\backslash rangle\; =\; \backslash left\backslash langle\; ^A\; (y\text{'}),\; x\; \backslash right\backslash rangle$ for all $x\; \backslash in\; X$ and all $y\text{'}\; \backslash in\; Y\text{'}.$ Since is continuous, the transpose $^A\; :\; Y\text{'}\; \backslash to\; X\text{'}$ 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 $A\; :\; \backslash mathcal(U)\; \backslash to\; \backslash mathcal(U)$ be a continuous linear map. Then by definition, the transpose of is the unique linear operator $A^t\; :\; \backslash mathcal\text{'}(U)\; \backslash to\; \backslash mathcal\text{'}(U)$ that satisfies:
:$\backslash langle\; ^A(T),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T,\; A(\backslash phi)\; \backslash rangle$ for all $\backslash phi\; \backslash in\; \backslash mathcal(U)$ and all $T\; \backslash in\; \backslash mathcal\text{'}(U).$
However, since the image of $\backslash mathcal(U)$ is dense in $\backslash mathcal\text{'}(U),$ it is sufficient that the above equality hold for all distributions of the form $T=\; D\_$ where $\backslash psi\; \backslash in\; \backslash mathcal(U).$ Explicitly, this means that the above condition holds if and only if the condition below holds:
:$\backslash langle\; ^A(D\_),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; D\_,\; A(\backslash phi)\; \backslash rangle\; =\; \backslash langle\; \backslash psi,\; A(\backslash phi)\; \backslash rangle\; =\; \backslash int\_U\; \backslash psi\; (A\backslash phi)\; \backslash ,dx$ for all $\backslash phi,\; \backslash psi\; \backslash in\; \backslash mathcal(U).$

** Differential operators **

** Differentiation of distributions **

Let $A:\backslash mathcal(U)\backslash to\; \backslash mathcal(U)$ is the partial derivative operator $\backslash tfrac.$ In order to extend $A$ we compute its transpose:
:$\backslash begin\; \backslash langle\; ^A(D\_),\; \backslash phi\; \backslash rangle\; \&=\; \backslash int\_U\; \backslash psi\; (A\backslash phi)\; \backslash ,dx\; \&\&\; \backslash text\; \backslash \backslash \; \&=\; \backslash int\_U\; \backslash psi\; \backslash frac\; \backslash ,\; dx\; \backslash \backslash pt\&=\; -\backslash int\_U\; \backslash phi\; \backslash frac\backslash ,\; dx\; \&\&\; \backslash text\; \backslash \backslash pt\&=\; -\backslash left\; \backslash langle\; \backslash frac,\; \backslash phi\; \backslash right\; \backslash rangle\; \backslash \backslash pt\&=\; -\backslash langle\; A\; \backslash psi,\; \backslash phi\; \backslash rangle\; \backslash end$
Therefore $^A=-A.$ Therefore the partial derivative of $T$ with respect to the coordinate $x\_k$ is defined by the formula
:$\backslash left\backslash langle\; \backslash frac,\; \backslash phi\; \backslash right\backslash rangle\; =\; -\; \backslash left\backslash langle\; T,\; \backslash frac\; \backslash right\backslash rangle\; \backslash qquad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(U).$
With this definition, every distribution is infinitely differentiable, and the derivative in the direction $x\_k$ is a linear operator on $\backslash mathcal\text{'}(U).$
More generally, if $\backslash alpha$ is an arbitrary multi-index, then the partial derivative $\backslash partial^T$ of the distribution $T\; \backslash in\; \backslash mathcal\text{'}(U)$ is defined by
:$\backslash langle\; \backslash partial^T,\; \backslash phi\; \backslash rangle\; =\; (-1)^\; \backslash langle\; T,\; \backslash partial^\; \backslash phi\; \backslash rangle\; \backslash qquad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(U).$
Differentiation of distributions is a continuous operator on $\backslash mathcal\text{'}(U);$ this is an important and desirable property that is not shared by most other notions of differentiation.
If is a distribution in then
:$\backslash lim\_\; \backslash frac\; =\; T\text{'}\backslash in\; \backslash mathcal\text{'}(\backslash R),$
where $T\text{'}$ is the derivative of and is 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 $U.$ Given $\backslash textstyle\; P\; :=\; \backslash sum\backslash nolimits\_\; c\_\; \backslash partial^$ we would like to define a continuous linear map, $D\_P$ that extends the action of $P$ on $C^(U)$ to distributions on $U.$ In other words we would like to define $D\_P$ such that the following diagram commutes:
:$\backslash begin\; \backslash mathcal\text{'}(U)\; \&\; \backslash stackrel\; \&\; \backslash mathcal\text{'}(U)\; \backslash \backslash \; \backslash uparrow\; \&\; \&\; \backslash uparrow\; \backslash \backslash \; C^(U)\; \&\; \backslash stackrel\; \&\; C^(U)\; \backslash end$
Where the vertical maps are given by assigning $f\; \backslash in\; C^(U)$ its canonical distribution $D\_f\; \backslash in\; \backslash mathcal\text{'}(U),$ which is defined by: $D\_f(\backslash phi)\; =\; \backslash langle\; f,\; \backslash phi\; \backslash rangle$ for all $\backslash phi\; \backslash in\; \backslash mathcal(U).$ With this notation the diagram commuting is equivalent to:
:$D\_\; =\; D\_PD\_f\; \backslash qquad\; \backslash text\; f\; \backslash in\; C^(U).$
In order to find $D\_P$ we consider the transpose $^P:\; \backslash mathcal\text{'}(U)\backslash to\; \backslash mathcal\text{'}(U)$ of the continuous induced map $P:\backslash mathcal(U)\backslash to\; \backslash mathcal(U)$ defined by $\backslash phi\; \backslash mapsto\; P(\backslash phi).$ As discussed above, for any $\backslash phi\; \backslash in\; \backslash mathcal(U),$ the transpose may be calculated by:
:$\backslash begin\; \backslash left\; \backslash langle\; ^P(D\_),\; \backslash phi\; \backslash right\; \backslash rangle\; \&=\; \backslash int\_U\; f(x)\; P(\backslash phi)(x)\; \backslash ,dx\; \backslash \backslash \; \&=\; \backslash int\_U\; f(x)\; \backslash leftsum\backslash nolimits\_\backslash alpha\; c\_(x)\; (\backslash partial^\; \backslash phi)(x)\; \backslash right\backslash ,dx\; \backslash \backslash \; \&=\; \backslash sum\backslash nolimits\_\backslash alpha\; \backslash int\_U\; f(x)\; c\_(x)\; (\backslash partial^\; \backslash phi)(x)\; \backslash ,dx\; \backslash \backslash \; \&=\; \backslash sum\backslash nolimits\_\backslash alpha\; (-1)^\; \backslash int\_U\; \backslash phi(x)\; (\backslash partial^(c\_f))(x)\; \backslash ,d\; x\; \backslash end$
For the last line we used integration by parts combined with the fact that $\backslash phi$ and therefore all the functions $f\; (x)c\_\; (x)\; \backslash partial^\; \backslash phi(x)$ have compact support.For example let $U\; =\; \backslash R$ and take $P$ to be the ordinary derivative for functions of one real variable and assume the support of $\backslash phi$ to be contained in the finite interval $(a,b),$ then since $\backslash operatorname(\backslash phi)\; \backslash subseteq\; (a,\; b)$
:$\backslash begin\; \backslash int\_\backslash phi\text{'}(x)f(x)\backslash ,dx\; \&=\; \backslash int\_a^b\; \backslash phi\text{'}(x)f(x)\; \backslash ,dx\; \backslash \backslash \; \&=\; \backslash phi(x)f(x)\backslash big\backslash vert\_^\; -\; \backslash int\_^\; f\text{'}(x)\; \backslash phi(x)\; \backslash ,d\; x\; \backslash \backslash \; \&=\; \backslash phi(b)f(b)\; -\; \backslash phi(a)f(a)\; -\; \backslash int\_^\; f\text{'}(x)\; \backslash phi(x)\; \backslash ,d\; x\; \backslash \backslash \; \&=\; \backslash int\_^\; f\text{'}(x)\; \backslash phi(x)\; \backslash ,d\; x\; \backslash end$
where the last equality is because $\backslash phi(a)\; =\; \backslash phi(b)\; =\; 0.$ Continuing the calculation above we have for all $\backslash phi\; \backslash in\; \backslash mathcal(U):$
:$\backslash begin\; \backslash left\; \backslash langle\; ^P(D\_),\; \backslash phi\; \backslash right\; \backslash rangle\; \&=\backslash sum\backslash nolimits\_\backslash alpha\; (-1)^\; \backslash int\_U\; \backslash phi(x)\; (\backslash partial^(c\_f))(x)\; \backslash ,dx\; \&\&\; \backslash text\; \backslash \backslash pt\&=\; \backslash int\_U\; \backslash phi(x)\; \backslash sum\backslash nolimits\_\backslash alpha\; (-1)^\; (\backslash partial^(c\_f))(x)\backslash ,dx\; \backslash \backslash pt\&=\; \backslash int\_U\; \backslash phi(x)\; \backslash sum\_\backslash alpha\; \backslash leftsum\_\; \backslash binom\; (\backslash partial^c\_)(x)\; (\backslash partial^f)(x)\; \backslash right\backslash ,dx\; \&\&\; \backslash text\backslash \backslash \; \&=\; \backslash int\_U\; \backslash phi(x)\; \backslash leftsum\_\backslash alpha\; \backslash sum\_\; (-1)^\; \backslash binom\; (\backslash partial^c\_)(x)\; (\backslash partial^f)(x)\backslash right\backslash ,dx\; \backslash \backslash \; \&=\; \backslash int\_U\; \backslash phi(x)\; \backslash left\backslash sum\_\backslash alpha\_\backslash left\_[\_\backslash sum\_\_(-1)^\_\backslash binom\_\backslash left(\backslash partial^c\_\backslash right)(x)\_\backslash right\_(\backslash partial^f)(x)\backslash right\_.html"\; style="text-decoration:\; none;"class="mw-redirect"\; title="\backslash sum\_\; (-1)^\; \backslash binom\; \backslash left(\backslash partial^c\_\backslash right)(x)\; \backslash right\; "\backslash sum\_\backslash alpha\; \backslash left\; [\; \backslash sum\_\; (-1)^\; \backslash binom\; \backslash left(\backslash partial^c\_\backslash right)(x)\; \backslash right\; (\backslash partial^f)(x)\backslash right\; "\backslash sum\_\; (-1)^\; \backslash binom\; \backslash left(\backslash partial^c\_\backslash right)(x)\; \backslash right\; "\backslash sum\_\backslash alpha\; \backslash left\; [\; \backslash sum\_\; (-1)^\; \backslash binom\; \backslash left(\backslash partial^c\_\backslash right)(x)\; \backslash right\; (\backslash partial^f)(x)\backslash right\; \backslash ,dx\; \backslash text\; f\; \backslash \backslash \; =\; \backslash int\_U\; \backslash phi(x)\; \backslash left\; [\backslash sum\backslash nolimits\_\backslash alpha\; b\_(x)\; (\backslash partial^f)(x)\; \backslash right\; ]\; \backslash ,\; dx\; b\_:=\backslash sum\_\; (-1)^\; \backslash binom\; \backslash partial^c\_\; \backslash \backslash \; =\; \backslash left\; \backslash langle\; \backslash left\; (\backslash sum\backslash nolimits\_\backslash alpha\; b\_\; \backslash partial^\; \backslash right\; )\; (f),\; \backslash phi\; \backslash right\; \backslash rangle\; \backslash end$
Define ''the formal transpose of $P,$'' which will be denoted by $P\_*$ to avoid confusion with the transpose map, to be the following differential operator on :
:$P\_*\; :=\; \backslash sum\backslash nolimits\_\backslash alpha\; b\_\; \backslash partial^$
The computations above have shown that:
:Lemma. Let $P$ be a linear differential operator with smooth coefficients in $U.$ Then for all $\backslash phi\; \backslash in\; \backslash mathcal(U)$ we have
::$\backslash left\; \backslash langle\; ^P(D\_),\; \backslash phi\; \backslash right\; \backslash rangle\; =\; \backslash left\; \backslash langle\; D\_,\; \backslash phi\; \backslash right\; \backslash rangle,$
:which is equivalent to:
::$^P(D\_)\; =\; D\_.$
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, i.e. $P\_=\; P,$ enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator $P\_*\; :\; C\_c^(U)\; \backslash to\; C\_c^(U)$ defined by $\backslash phi\; \backslash mapsto\; P\_*(\backslash phi).$ We claim that the transpose of this map, $^P\_*\; :\; \backslash mathcal\text{'}(U)\; \backslash to\; \backslash mathcal\text{'}(U),$ can be taken as $D\_P.$ To see this, for every $\backslash phi\; \backslash in\; \backslash mathcal(U)$, compute its action on a distribution of the form $D\_f$ with $f\; \backslash in\; C^(U)$:
:$\backslash begin\; \backslash left\; \backslash langle\; ^P\_*(D\_f),\backslash phi\; \backslash right\; \backslash rangle\; \&=\; \backslash left\; \backslash langle\; D\_,\; \backslash phi\; \backslash right\; \backslash rangle\; \&\&\; \backslash text\; P\_*\; \backslash text\; P\backslash \backslash \; \&=\; \backslash left\; \backslash langle\; D\_,\; \backslash phi\; \backslash right\; \backslash rangle\; \&\&\; P\_=\; P\; \backslash end$
We call the continuous linear operator $D\_P:=^P\_*\; :\; \backslash mathcal\text{'}(U)\; \backslash to\; \backslash mathcal\text{'}(U)$ ''the differential operator on distributions extending ''. Its action on an arbitrary distribution $S$ is defined via:
:$D\_(S)(\backslash phi)\; =\; S(P\_*(\backslash phi))\; \backslash quad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(U).$
If $(T\_i)\_^$ converges to $T\; \backslash in\; \backslash mathcal\text{'}(U)$ then for every multi-index $\backslash alpha,\; (\backslash partial^T\_i)\_^$ converges to $\backslash partial^T\; \backslash in\; \backslash mathcal\text{'}(U).$

** Multiplication of distributions by smooth functions **

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if $f$ is a smooth function then $P\; :=\; f(x)$ is a differential operator of order 0, whose formal transpose is itself (i.e. $P\_*\; =\; P$). The induced differential operator $D\_:\; \backslash mathcal\text{'}(U)\; \backslash to\; \backslash mathcal\text{'}(U)$ maps a distribution to a distribution denoted by $fT\; :=\; D\_(T).$ 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 $m:\; U\; \backslash to\; \backslash R$ is a smooth function and is a distribution on , then the product ''mT'' is defined by
:$\backslash langle\; mT,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T,\; m\backslash phi\; \backslash rangle\; \backslash qquad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(U).$
This definition coincides with the transpose definition since if $M:\backslash mathcal(U)\backslash to\backslash mathcal(U)$ is the operator of multiplication by the function (i.e., $(M\backslash phi)(x)=m(x)\backslash phi(x)$), then
:$\backslash int\_U\; (M\; \backslash phi)(x)\; \backslash psi(x)\backslash ,dx\; =\; \backslash int\_U\; m(x)\; \backslash phi(x)\; \backslash psi(x)\backslash ,d\; x\; =\; \backslash int\_U\; \backslash phi(x)\; m(x)\; \backslash psi(x)\; \backslash ,d\; x\; =\; \backslash int\_U\; \backslash phi(x)\; (M\; \backslash psi)(x)\backslash ,d\; x,$
so that $^tM=M.$
Under multiplication by smooth functions, $\backslash mathcal\text{'}(U)$ is a module over the ring $C^\backslash infty(U).$ 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
:$m\backslash delta\text{'}\; =\; m(0)\; \backslash delta\text{'}\; -\; m\text{'}\; \backslash delta\; =\; m(0)\; \backslash delta\text{'}\; -\; m\text{'}(0)\; \backslash delta.$
The bilinear multiplication map $C^(\backslash R^n)\; \backslash times\; \backslash mathcal\text{'}(\backslash R^n)\; \backslash to\; \backslash mathcal\text{'}(\backslash R^n)$ given by $(f,T)\backslash mapsto\; fT$ is ''not'' continuous; it is however, hypocontinuous.
Example. For any distribution , the product of with the function that is identically on is equal to .
Example. Suppose $(f\_i)\_^$ is a sequence of test functions on that converges to the constant function $1\; \backslash in\; C^(U).$ For any distribution on , the sequence $(f\_i\; T)\_^$ converges to $T\; \backslash in\; \backslash mathcal\text{'}(U).$
If $(T\_i)\_^$ converges to $T\; \backslash in\; \backslash mathcal\text{'}(U)$ and $(f\_i)\_^$ converges to $f\; \backslash in\; C^(U)$ then $(f\_i\; T\_i)\_^$ converges to $fT\; \backslash in\; \backslash mathcal\text{'}(U).$

= Problem of multiplying distributions

= It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if p.v. is the distribution obtained by the Cauchy principal value :$\backslash left(\backslash operatorname\backslash frac\backslash right)(\backslash phi)\; =\; \backslash lim\_\; \backslash int\_\; \backslash frac\backslash ,\; dx\; \backslash quad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(\backslash R).$ If is the Dirac delta distribution then :$(\backslash delta\; \backslash times\; x)\; \backslash times\; \backslash operatorname\; \backslash frac\; =\; 0$ but :$\backslash delta\; \backslash times\; \backslash left(x\; \backslash times\; \backslash operatorname\; \backslash frac\backslash right)\; =\; \backslash delta$ 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. 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 be a distribution on $U.$ Let be an open set in $\backslash R^n,$ and . If is a submersion, it is possible to define
:$T\; \backslash circ\; F\; \backslash in\; \backslash mathcal\text{'}(V).$
This is ''the composition of the distribution with '', and is also called ''the pullback of along '', sometimes written
:$F^\backslash sharp\; :\; T\; \backslash mapsto\; F^\backslash sharp\; T\; =\; T\; \backslash circ\; F.$
The pullback is often denoted ''F*'', 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. Uniqueness of this extension is guaranteed since is a continuous linear operator on $\backslash mathcal(U).$ 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 $\backslash R^n$ onto an open subset of $\backslash R^n$ change of variables under the integral gives
:$\backslash int\_V\; \backslash phi\backslash circ\; F(x)\; \backslash psi(x)\backslash ,dx\; =\; \backslash int\_U\; \backslash phi(x)\; \backslash psi\; \backslash left\; (F^(x)\; \backslash right\; )\; \backslash left\; |\backslash det\; dF^(x)\; \backslash right\; |\backslash ,dx.$
In this particular case, then, is defined by the transpose formula:
:$\backslash left\; \backslash langle\; F^\backslash sharp\; T,\; \backslash phi\; \backslash right\; \backslash rangle\; =\; \backslash left\; \backslash langle\; T,\; \backslash left\; |\backslash det\; d(F^)\; \backslash right\; |\; \backslash phi\backslash circ\; F^\; \backslash right\; \backslash rangle.$

** Convolution **

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 $f$ and are functions on $\backslash R^n$ then we denote by $f\backslash ast\; g$ ''the convolution of $f$ and '', defined at $x\; \backslash in\; \backslash R^n$ to be the integral
:$(f\; \backslash ast\; g)(x)\; :=\; \backslash int\_\; f(x-y)\; g(y)\; \backslash ,dy\; =\; \backslash int\_\; f(y)g(x-y)\; \backslash ,dy$
provided that the integral exists. If $1\; \backslash leq\; p,\; q,\; r\; \backslash leq\; \backslash infty$ are such that 1/''r'' = (1/''p'') + (1/''q'') - 1 then for any functions $f\; \backslash in\; L^p(\backslash R^n)$ and $g\; \backslash in\; L^q(\backslash R^n)$ we have $f\; \backslash ast\; g\; \backslash in\; L^r(\backslash R^n)$ and $\backslash |f\; \backslash ast\; g\backslash |\_\; \backslash leq\; \backslash |\; f\backslash |\_\; \backslash |\; g\backslash |\_.$ If $f$ and are continuous functions on $\backslash R^n,$ at least one of which has compact support, then $\backslash operatorname(f\; \backslash ast\; g)\; \backslash subseteq\; \backslash operatorname\; (f)\; +\; \backslash operatorname\; (g)$ and if $A\backslash subseteq\; \backslash R^n$ then the value of $f\backslash ast\; g$ on do ''not'' depend on the values of $f$ outside of the Minkowski sum $A\; -\backslash operatorname\; (g)\; =\; \backslash .$
Importantly, if $g\; \backslash in\; L^1(\backslash R^n)$ has compact support then for any $0\; \backslash leq\; k\; \backslash leq\; \backslash infty,$ the convolution map $f\; \backslash mapsto\; f\; \backslash ast\; g$ is continuous when considered as the map $C^k(\backslash R^n)\; \backslash to\; C^k(\backslash R^n)$ or as the map $C\_c^k(\backslash R^n)\; \backslash to\; C\_c^k(\backslash R^n).$
;Translation and symmetry
Given $a\; \backslash in\; \backslash R^n,$ the translation operator sends $f\; :\; \backslash R^n\; \backslash to\; \backslash Complex$ to $\backslash tau\_a\; f\; :\; \backslash R^n\; \backslash to\; \backslash Complex,$ defined by $\backslash tau\_a\; f(y)\; =\; f(y-a).$ This can be extended by the transpose to distributions in the following way: given a distribution , ''the translation of $T$ by $a$'' is the distribution $\backslash tau\_a\; T\; :\; \backslash mathcal(\backslash R^n)\; \backslash to\; \backslash Complex$ defined by $\backslash tau\_a\; T(\backslash phi)\; :=\; \backslash left\backslash langle\; T,\; \backslash tau\_\; \backslash phi\; \backslash right\backslash rangle.$
Given $f\; :\; \backslash R^n\; \backslash to\; \backslash Complex,$ define the function $\backslash tilde\; :\; \backslash R^n\; \backslash to\; \backslash Complex$ by $\backslash tilde(x)\; :=\; f(-x).$ Given a distribution , let $\backslash tilde\; :\; \backslash mathcal(\backslash R^n)\; \backslash to\; \backslash Complex$ be the distribution defined by $\backslash tilde(\backslash phi)\; :=\; T\; \backslash left(\backslash tilde\backslash right).$ The operator $T\; \backslash mapsto\; \backslash tilde$ is called ''the symmetry with respect to the origin''.

** Convolution of a test function with a distribution **

Convolution with $f\; \backslash in\; \backslash mathcal(\backslash R^n)$ defines a linear map:
:$\backslash begin\; C\_f\; :\; \backslash mathcal(\backslash R^n)\; \backslash to\; \backslash mathcal(\backslash R^n)\; \backslash \backslash \; C\_f(g)\; :=\; f\; \backslash ast\; g\; \backslash end$
which is continuous with respect to the canonical LF space topology on $\backslash mathcal(\backslash R^n).$
Convolution of $f$ with a distribution $T\; \backslash in\; \backslash mathcal\text{'}(\backslash R^n)$ can be defined by taking the transpose of ''C_{f}'' relative to the duality pairing of $\backslash mathcal(\backslash R^n)$ with the space $\backslash mathcal\text{'}(\backslash R^n)$ of distributions. If $f,\; g,\; \backslash phi\; \backslash in\; \backslash mathcal(\backslash R^n),$ then by Fubini's theorem
:$\backslash langle\; C\_fg,\; \backslash phi\; \backslash rangle\; =\; \backslash int\_\backslash phi(x)\backslash int\_f(x-y)\; g(y)\; \backslash ,dy\; \backslash ,dx\; =\; \backslash left\; \backslash langle\; g,C\_\backslash phi\; \backslash right\; \backslash rangle.$
Extending by continuity, the convolution of $f$ with a distribution is defined by
:$\backslash langle\; f\; \backslash ast\; T,\; \backslash phi\; \backslash rangle\; =\; \backslash left\; \backslash langle\; T,\; \backslash tilde\; \backslash ast\; \backslash phi\; \backslash right\; \backslash rangle,$
for all $\backslash phi\; \backslash in\; \backslash mathcal(\backslash R^n).$
An alternative way to define the convolution of a test function $f$ and a distribution is to use the translation operator . The convolution of the compactly supported function $f$ and the distribution is then the function defined for each $x\; \backslash in\; \backslash R^n$ by
:$(f\; \backslash ast\; T)(x)\; =\; \backslash left\; \backslash langle\; T,\; \backslash tau\_x\; \backslash tilde\; \backslash right\; \backslash rangle.$
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 then if $f$ is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on $\backslash Complex^n$ to $\backslash R^n,$ the restriction of an entire function of exponential type in $\backslash Complex^n$ to $\backslash R^n$) then the same is true of $T\; \backslash ast\; f.$ If the distribution has compact support as well, then $f\backslash ast\; T$ is a compactly supported function, and the Titchmarsh convolution theorem implies that
:$\backslash operatorname(\backslash operatorname(f\; \backslash ast\; T))\; =\; \backslash operatorname(\backslash operatorname(f))\; +\; \backslash operatorname\; (\backslash operatorname(T))$
where ''ch'' denotes the convex hull and supp denotes the support.

** Convolution of a smooth function with a distribution **

Let $f\; \backslash in\; C^(\backslash R^n)$ and $T\; \backslash in\; \backslash mathcal\text{'}(\backslash R^n)$ and assume that at least one of $f$ and has compact support. The ''convolution of $f$ and '', denoted by $f\; \backslash ast\; T$ or by $T\; \backslash ast\; f,$ is the smooth function:
:$\backslash beginf\; \backslash ast\; T:\; \backslash R^n\; \backslash to\; \backslash Complex\; \backslash \backslash \; (f\; \backslash ast\; T)(x)\; :=\; \backslash left\backslash langle\; T,\; \backslash tau\_\backslash tilde\; \backslash right\backslash rangle\backslash end$
satisfying for all $p\; \backslash in\; \backslash N^n$:
:$\backslash begin\; \&\backslash operatorname(f\; \backslash ast\; T)\; \backslash subseteq\; \backslash operatorname(f)+\; \backslash operatorname(T)\; \backslash \backslash pt\&\backslash text\; p\; \backslash in\; \backslash N^n:\; \backslash quad\; \backslash begin\backslash partial^\; \backslash left\backslash langle\; T,\; \backslash tau\_x\; \backslash tilde\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; T,\; \backslash partial^\; \backslash tau\_\; \backslash tilde\; \backslash right\backslash rangle\; \backslash \backslash \; \backslash partial^\; (T\; \backslash ast\; f)\; =\; (\backslash partial^\; T)\; \backslash ast\; f\; =\; T\; \backslash ast\; (\backslash partial^\; f)\; \backslash end.\; \backslash end$
If is a distribution then the map $f\; \backslash mapsto\; T\; \backslash ast\; f$ is continuous as a map $\backslash mathcal(\backslash R^n)\; \backslash to\; C^(\backslash R^n)$ where if in addition has compact support then it is also continuous as the map $C^(\backslash R^n)\; \backslash to\; C^(\backslash R^n)$ and continuous as the map $\backslash mathcal(\backslash R^n)\; \backslash to\; \backslash mathcal(\backslash R^n).$
If $L\; :\; \backslash mathcal(\backslash R^n)\; \backslash to\; C^(\backslash R^n)$ is a continuous linear map such that $L\; \backslash partial^\; \backslash phi\; =\; \backslash partial^L\; \backslash phi$ for all $\backslash alpha$ and all $\backslash phi\; \backslash in\; \backslash mathcal(\backslash R^n)$ then there exists a distribution $T\; \backslash in\; \backslash mathcal\text{'}(\backslash R^n)$ such that $L\; \backslash phi\; =\; T\; \backslash circ\; \backslash phi$ for all $\backslash phi\; \backslash in\; \backslash mathcal(\backslash R^n).$
Example. Let ''H'' be the Heaviside function on . For any $\backslash phi\; \backslash in\; \backslash mathcal(\backslash R),$
:$(H\; \backslash ast\; \backslash phi)(x)\; =\; \backslash int\_^\; \backslash phi(t)\; \backslash ,\; dt.$
Let $\backslash delta$ be the Dirac measure at 0 and $\backslash delta\text{'}$ its derivative as a distribution. Then $\backslash delta\text{'}\; \backslash ast\; H\; =\; \backslash delta$ and $1\; \backslash ast\; \backslash delta\text{'}\; =\; 0.$ Importantly, the associative law fails to hold:
:$1\; =\; 1\; \backslash ast\; \backslash delta\; =\; 1\; \backslash ast\; (\backslash delta\text{'}\; \backslash ast\; H\; )\; \backslash neq\; (1\; \backslash ast\; \backslash delta\text{'})\; \backslash ast\; H\; =\; 0\; \backslash ast\; H\; =\; 0.$

** Convolution of distributions **

It is also possible to define the convolution of two distributions and on $\backslash R^n,$ provided one of them has compact support. Informally, in order to define where has compact support, the idea is to extend the definition of the convolution to a linear operation on distributions so that the associativity formula
:$S\; \backslash ast\; (T\; \backslash ast\; \backslash phi)\; =\; (S\; \backslash ast\; T)\; \backslash ast\; \backslash phi$
continues to hold for all test functions $\backslash phi.$
It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that and are distributions and that has compact support. Then the linear maps
:$\backslash begin\; \backslash bullet\; \backslash ast\; \backslash tilde\; :\; \backslash mathcal(\backslash R^n)\; \backslash to\; \backslash mathcal(\backslash R^n)\; \backslash \backslash \; f\; \backslash mapsto\; f\; \backslash ast\; \backslash tilde\backslash end\; \backslash qquad\; \backslash begin\; \backslash bullet\; \backslash ast\; \backslash tilde\; :\; \backslash mathcal(\backslash R^n)\; \backslash to\; C^(\backslash R^n)\backslash \backslash \; f\; \backslash mapsto\; f\; \backslash ast\; \backslash tilde\backslash end$
are continuous. The transposes of these maps,
:$^\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\backslash right)\; :\; \backslash mathcal\text{'}(\backslash R^n)\; \backslash to\; \backslash mathcal\text{'}(\backslash R^n)\; \backslash qquad\; ^\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\backslash right)\; :\; \backslash mathcal\text{'}(\backslash R^n)\; \backslash to\; \backslash mathcal\text{'}(\backslash R^n)$
are consequently continuous and one may show that
:$^\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\backslash right)(T)\; =\; ^\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\backslash right)(S).$
This common value is called ''the convolution of and '' and it is a distribution that is denoted by $S\; \backslash ast\; T$ or $T\; \backslash ast\; S.$ It satisfies $\backslash operatorname\; (S\; \backslash ast\; T)\; \backslash subseteq\; \backslash operatorname(S)\; +\; \backslash operatorname(T).$ If and are two distributions, at least one of which has compact support, then for any $a\; \backslash in\; \backslash R^n,$ $\backslash tau\_a(S\; \backslash ast\; T)\; =\; \backslash left(\backslash tau\_a\; S\backslash right)\; \backslash ast\; T\; =\; S\; \backslash ast\; \backslash left(\backslash tau\_a\; T\backslash right).$ If is a distribution in $\backslash R^n$ and if $\backslash delta$ is a Dirac measure then $T\; \backslash ast\; \backslash delta\; =\; T.$
Suppose that it is that has compact support. For $\backslash phi\; \backslash in\; \backslash mathcal(\backslash R^n)$ consider the function
:$\backslash psi(x)\; =\; \backslash langle\; T,\; \backslash tau\_\; \backslash phi\; \backslash rangle.$
It can be readily shown that this defines a smooth function of , which moreover has compact support. The convolution of and is defined by
:$\backslash langle\; S\; \backslash ast\; T,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; S,\; \backslash psi\; \backslash rangle.$
This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index ,
:$\backslash partial^\backslash alpha(S\; \backslash ast\; T)\; =\; (\backslash partial^\backslash alpha\; S)\; \backslash ast\; T\; =\; S\; \backslash ast\; (\backslash partial^\backslash alpha\; T).$
The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.
This definition of convolution remains valid under less restrictive assumptions about and .
The convolution of distributions with compact support induces a continuous bilinear map $\backslash mathcal\text{'}\; \backslash times\; \backslash mathcal\text{'}\; \backslash to\; \backslash mathcal\text{'}$ defined by $(S,T)\; \backslash mapsto\; S\; *\; T,$ where $\backslash mathcal\text{'}$ denotes the space of distributions with compact support. However, the convolution map as a function $\backslash mathcal\text{'}\; \backslash times\; \backslash mathcal\text{'}\; \backslash to\; \backslash mathcal\text{'}$ is ''not'' continuous although it is separately continuous. The convolution maps $\backslash mathcal(\backslash R^n)\; \backslash times\; \backslash mathcal\text{'}\; \backslash to\; \backslash mathcal\text{'}$ and $\backslash mathcal(\backslash R^n)\; \backslash times\; \backslash mathcal\text{'}\; \backslash to\; \backslash mathcal(\backslash R^n)$ given by $(f,\; T)\; \backslash mapsto\; f\; *\; T$ 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 $F(\backslash alpha)\; =\; f\; \backslash in\; \backslash mathcal\text{'}\_C$ be a rapidly decreasing tempered distribution or, equivalently, $F(f)\; =\; \backslash alpha\; \backslash in\; \backslash mathcal\_M$ be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let $F$ be the normalized (unitary, ordinary frequency) Fourier transform then, according to ,
:$F(f\; *\; g)\; =\; F(f)\; \backslash cdot\; F(g)$
:$F(\backslash alpha\; \backslash cdot\; g)\; =\; F(\backslash alpha)\; *\; F(g)$
hold within the space of tempered distributions. In particular, these equations become the Poisson Summation Formula if $g\; \backslash equiv\; \backslash operatorname$ is the Dirac Comb. The space of all rapidly decreasing tempered distributions is also called the space of ''convolution operators'' $\backslash mathcal\text{'}\_C$ and the space of all ordinary functions within the space of tempered distributions is also called the space of ''multiplication operators'' $\backslash mathcal\_M.$ More generally, $F(\backslash mathcal\text{'}\_C)\; =\; \backslash mathcal\_M$ and $F(\backslash mathcal\_M)\; =\; \backslash mathcal\text{'}\_C.$ A particular case is the Paley-Wiener-Schwartz Theorem which states that $F(\backslash mathcal\text{'})\; =\; \backslash operatorname$ and $F(\backslash operatorname\; )\; =\; \backslash mathcal\text{'}.$ This is because $\backslash mathcal\text{'}\; \backslash subseteq\; \backslash mathcal\text{'}\_C$ and $\backslash operatorname\; \backslash subseteq\; \backslash mathcal\_M.$ In other words, compactly supported tempered distributions $\backslash mathcal\text{'}$ belong to the space of ''convolution operators'' $\backslash mathcal\text{'}\_C$ and
Paley-Wiener functions $\backslash operatorname,$ better known as bandlimited functions, belong to the space of ''multiplication operators'' $\backslash mathcal\_M.$
For example, let $g\; \backslash equiv\; \backslash operatorname\; \backslash in\; \backslash mathcal\text{'}$ be the Dirac comb and $f\; \backslash equiv\; \backslash delta\; \backslash in\; \backslash mathcal\text{'}$ be the Dirac delta then $\backslash alpha\; \backslash equiv\; 1\; \backslash in\; \backslash operatorname$ is the function that is constantly one and both equations yield the Dirac comb identity. Another example is to let $g$ be the Dirac comb and $f\; \backslash equiv\; \backslash operatorname\; \backslash in\; \backslash mathcal\text{'}$ be the rectangular function then $\backslash alpha\; \backslash equiv\; \backslash operatorname\; \backslash in\; \backslash operatorname$ is the sinc function and both equations yield the Classical Sampling Theorem for suitable $\backslash operatorname$ functions. More generally, if $g$ is the Dirac comb and $f\; \backslash in\; \backslash mathcal\; \backslash subseteq\; \backslash mathcal\text{'}\_C\; \backslash cap\; \backslash mathcal\_M$ is a smooth window function (Schwartz function), e.g. the Gaussian, then $\backslash alpha\; \backslash in\; \backslash mathcal$ 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 $U\backslash subseteq\; \backslash R^m$ and $V\backslash subseteq\; \backslash R^n$ be open sets. Assume all vector spaces to be over the field $\backslash mathbb,$ where $\backslash mathbb=\backslash R$ or $\backslash Complex.$ For $f\; \backslash in\; \backslash mathcal(U\; \backslash times\; V)$ we define the following family of functions:
:$\backslash left\; \backslash ,\; \backslash qquad\; \backslash left\; \backslash .$
Given $S\; \backslash in\; \backslash mathcal\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\text{'}(V)$ we define the following functions:
:$\backslash begin\; \backslash begin\; \backslash langle\; S,\; f^\backslash rangle\; :\; V\; \backslash to\; \backslash mathbb\; \backslash \backslash \; y\; \backslash mapsto\; \backslash langle\; S,\; f^\; \backslash rangle\; \backslash end\; \backslash \backslash pt\backslash begin\; \backslash langle\; T,\; f\_\backslash rangle\; :\; U\; \backslash to\; \backslash mathbb\; \backslash \backslash \; x\; \backslash mapsto\; \backslash langle\; T,\; f\_\; \backslash rangle\; \backslash end\; \backslash end$
Note that $\backslash langle\; T,\; f\_\backslash rangle\; \backslash in\; \backslash mathcal(U)$ and $\backslash langle\; S,\; f^\backslash rangle\; \backslash in\; \backslash mathcal(V).$ Now we define the following continuous linear maps associated to $S$ and $T$:
:$\backslash begin\; \backslash mathcal\text{'}(U)\; \backslash ni\; S\; \&\backslash longrightarrow\; \backslash begin\; \backslash mathcal(U\; \backslash times\; V)\; \backslash to\; \backslash mathcal(V)\; \backslash \backslash \; f\; \backslash mapsto\; \backslash langle\; S,\; f^\; \backslash rangle\backslash end\; \backslash \backslash pt\backslash mathcal\text{'}(V)\; \backslash ni\; T\; \&\backslash longrightarrow\; \backslash begin\; \backslash mathcal(U\; \backslash times\; V)\; \backslash to\; \backslash mathcal(U)\; \backslash \backslash \; f\; \backslash mapsto\; \backslash langle\; T,\; f\_\; \backslash rangle\backslash end\; \backslash end$
Moreover if either $S$ (resp. $T$) has compact support then it also induces a continuous linear map of $C^(U\; \backslash times\; V)\; \backslash to\; C^(V)$ (resp. $C^(U\; \backslash times\; V)\; \backslash to\; C^(U)$).
Definition. ''The tensor product of $S\; \backslash in\; \backslash mathcal\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\text{'}(V),$'' denoted by $S\; \backslash otimes\; T$ or $T\; \backslash otimes\; S,$ is a distribution in $U\; \backslash times\; V$ and is defined by:
:$(S\; \backslash otimes\; T)(f)\; :=\; \backslash langle\; S,\; \backslash langle\; T,\; f\_\; \backslash rangle\; \backslash rangle\; =\; \backslash langle\; T,\; \backslash langle\; S,\; f^\backslash rangle\; \backslash rangle.$

** Schwartz kernel theorem **

The tensor product defines a bilinear map
:$\backslash begin\backslash mathcal\text{'}(U)\; \backslash times\; \backslash mathcal\text{'}(V)\; \backslash to\; \backslash mathcal\text{'}(U\; \backslash times\; V)\; \backslash \backslash \; (S,T)\; \backslash mapsto\; S\; \backslash otimes\; T\backslash end$
the span of the range of this map is a dense subspace of its codomain. Furthermore, $\backslash operatorname\; (S\; \backslash otimes\; T)\; =\; \backslash operatorname(S)\; \backslash times\; \backslash operatorname(T).$ Moreover $(S,T)\; \backslash mapsto\; S\; \backslash otimes\; T$ induces continuous bilinear maps:
:$\backslash begin\; \backslash mathcal\text{'}(U)\; \backslash times\; \backslash mathcal\text{'}(V)\; \&\backslash to\; \backslash mathcal\text{'}(U\; \backslash times\; V)\; \backslash \backslash \; \backslash mathcal\text{'}(\backslash R^m)\; \backslash times\; \backslash mathcal\text{'}(\backslash R^n)\; \&\backslash to\; \backslash mathcal\text{'}(\backslash R^)\; \backslash end$
where $\backslash mathcal\text{'}$ denotes the space of distributions with compact support and $\backslash mathcal$ is the Schwartz space of rapidly decreasing functions.
This result does not hold for Hilbert spaces such as $L^2$ 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 $L^2$? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because $\backslash mathcal(U)$ is a nuclear space that the Schwartz kernel theorem holds.

** Spaces of distributions **

For all and all , all of the following canonical injections are continuous and have a range that is dense in their codomain:
:$\backslash begin\; C\_c^(U)\; \&\; \backslash to\; \&\; C\_c^k(U)\; \&\; \backslash to\; \&\; C\_c^0(U)\; \&\; \backslash to\; \&\; L\_c^(U)\; \&\; \backslash to\; \&\; L\_c^p(U)\; \&\; \backslash to\; \&\; L\_c^1(U)\backslash \backslash \; \backslash downarrow\; \&\; \&\backslash downarrow\; \&\&\; \backslash downarrow\; \&\&\; \&\&\; \&\&\; \backslash \backslash \; C^(U)\; \&\; \backslash to\; \&\; C^k(U)\; \&\; \backslash to\; \&\; C^0(U)\; \&\&\; \&\&\; \&\&\; \backslash \backslash \; \backslash end$
where the topologies on $L\_c^(U)$ ($1\; \backslash leq\; q\; \backslash leq\; \backslash infty$) are defined as direct limits of the spaces $L\_c^q(K)$ in a manner analogous to how the topologies on $C\_c^k(U)$ were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in the codomain. Indeed, $C\_c^(U)$ is even sequentially dense in every $C\_c^k(U).$ All of the canonical injections $C\_c^(U)\; \backslash to\; L^p(U)$ ($1\; \backslash leq\; p\; \backslash leq\; \backslash infty$) are continuous and the range of this injection is dense in the codomain if and only if $p\; \backslash neq\; \backslash infty$ (here $L^p(U)$ has its usual norm topology).
Suppose that $X$ is one of the spaces $C\_c^k(U)$ ($k\; \backslash in\; \backslash $) or $L^p\_c(U)$ ($1\; \backslash leq\; p\; \backslash leq\; \backslash infty$) or $L^p(U)$ ($1\; \backslash leq\; p\; <\; \backslash infty$). Since the canonical injection $\backslash operatorname\_X\; :\; C\_c^(U)\; \backslash to\; X$ is a continuous injection whose image is dense in the codomain, the transpose $^\backslash operatorname\_X\; :\; X\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(U)\; =\; (C\_c^(U))\text{'}\_b$ is a continuous injection. This transpose thus allows us to identify $X\text{'}$ with a certain vector subspace of the space of distributions. This transpose map is not necessarily a TVS-embedding so that topology that this map transfers to the image $\backslash operatorname\backslash left(^\backslash operatorname\_X\backslash right)$ is finer than the subspace topology that this space inherits from $\backslash mathcal\text{'}(U).$
A linear subspace of $\backslash mathcal\text{'}(U)$ carrying a locally convex topology that is finer than the subspace topology induced by $\backslash mathcal\text{'}(U)\; =\; (C\_c^(U))\text{'}\_b$ is called ''a space of distributions''.
Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order $\backslash leq$ some integer, distributions induced by a positive Radon measure, distributions induced by an $L^p$-function, etc.) and any representation theorem about the dual space of may, through the transpose $^\backslash operatorname\_X\; :\; X\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(U),$ be transferred directly to elements of the space $\backslash operatorname\; \backslash left(^\backslash operatorname\_X\backslash right).$

** Radon measures **

The natural inclusion $\backslash operatorname\; :\; C\_c^(U)\; \backslash to\; C\_c^0(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $^\backslash operatorname\; :\; (C\_c^0(U))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(U)\; =\; (C\_c^(U))\text{'}\_b$ is also a continuous injection.
Note that the continuous dual space $(C\_c^0(U))\text{'}\_b$ can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals $T\; \backslash in\; (C\_c^0(U))\text{'}\_b$ and integral with respect to a Radon measure; that is,
* if $T\; \backslash in\; (C\_c^0(U))\text{'}\_b$ then there exists a Radon measure $\backslash mu$ on such that for all $f\; \backslash in\; C\_c^0(U),\; T(f)\; =\; \backslash textstyle\; \backslash int\_\; f\; \backslash ,\; d\backslash mu,$ and
* if $\backslash mu$ is a Radon measure on then the linear functional on $C\_c^0(U)$ defined by $C\_c^0(U)\; \backslash ni\; f\; \backslash mapsto\; \backslash textstyle\; \backslash int\_\; f\; \backslash ,\; d\backslash mu$ is continuous.
Through the injection $^\backslash operatorname\; :\; (C\_c^0(U))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(U),$ every Radon measure becomes a distribution on . If $f$ is a locally integrable function on then the distribution $\backslash phi\; \backslash mapsto\; \backslash textstyle\; \backslash int\_U\; f(x)\; \backslash phi(x)\; \backslash ,\; dx$ 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 $L^$ functions in :
:Theorem. Suppose $T\; \backslash in\; \backslash mathcal\text{'}(U)$ is a Radon measure, is a neighborhood of the support of , and $I\; =\; \backslash .$ There exists is a family of locally $L^$ functions in such that
::$T\; =\; \backslash sum\_\; \backslash partial^p\; \backslash mu\_p$
:and for very $p\; \backslash in\; I,\; \backslash operatorname\; \backslash mu\_p\; \backslash subseteq\; V.$
;Positive Radon measures
A linear function on a space of functions is called ''positive'' if whenever a function $f$ that belongs to the domain of is non-negative (i.e. $f$ is real-valued and $f\; \backslash geq\; 0$) then $T(\; f\; )\; \backslash geq\; 0.$ One may show that every positive linear functional on $C\_c^0(U)$ is necessarily continuous (i.e. necessarily a Radon measure).Note that 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 $f:U\backslash to\backslash R$ is called ''locally integrable'' if it is Lebesgue integrable over every compact subset of .For more information on such class of functions, see the entry on locally integrable functions. This is a large class of functions which includes all continuous functions and all ''L^{p}'' functions. The topology on $\backslash mathcal(U)$ is defined in such a fashion that any locally integrable function $f$ yields a continuous linear functional on $\backslash mathcal(U)$ – that is, an element of $\backslash mathcal\text{'}(U)$ – denoted here by , whose value on the test function $\backslash phi$ is given by the Lebesgue integral:
:$\backslash langle\; T\_f,\; \backslash phi\; \backslash rangle\; =\; \backslash int\_U\; f\; \backslash phi\backslash ,dx.$
Conventionally, one abuses notation by identifying with $f,$ provided no confusion can arise, and thus the pairing between and $\backslash phi$ is often written
:$\backslash langle\; f,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T\_f,\; \backslash phi\; \backslash rangle.$
If $f$ and are two locally integrable functions, then the associated distributions and are equal to the same element of $\backslash mathcal\text{'}(U)$ if and only if $f$ and are equal almost everywhere (see, for instance, ). In a similar manner, every Radon measure $\backslash mu$ on defines an element of $\backslash mathcal\text{'}(U)$ whose value on the test function $\backslash phi$ is $\backslash textstyle\backslash int\backslash phi\; \backslash ,d\backslash mu.$ As above, it is conventional to abuse notation and write the pairing between a Radon measure $\backslash mu$ and a test function $\backslash phi$ as $\backslash langle\; \backslash mu,\; \backslash phi\; \backslash rangle.$ Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.
;Test functions as distributions
The test functions are themselves locally integrable, and so define distributions. The space of test functions $C\_c^(U)$ is sequentially dense in $\backslash mathcal\text{'}(U)$ with respect to the strong topology on $\backslash mathcal\text{'}(U).$ This means that for any $T\; \backslash in\; \backslash mathcal\text{'}(U),$ there is a sequence of test functions, $(\backslash phi\_i)\_^,$ that converges to $T\; \backslash in\; \backslash mathcal\text{'}(U)$ (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
:$\backslash langle\; \backslash phi\_i,\; \backslash psi\; \backslash rangle\; \backslash to\; \backslash langle\; T,\; \backslash psi\; \backslash rangle\; \backslash qquad\; \backslash text\; \backslash psi\; \backslash in\; \backslash mathcal(U).$
Furthermore, $C\_c^(U)$ is also sequentially dense in the strong dual space of $C^(U).$

** Distributions with compact support **

The natural inclusion $\backslash operatorname:\; C\_c^(U)\; \backslash to\; C^(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $^\backslash operatorname:\; (C^(U))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(U)\; =\; (C\_c^(U))\text{'}\_b$ is also a continuous injection. Thus the image of the transpose, denoted by $\backslash mathcal\text{'}(U),$ forms a space of distributions when it is endowed with the strong dual topology of $(C^(U))\text{'}\_b$ (transferred to it via the transpose map $^\backslash operatorname:\; (C^(U))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(U),$ so the topology of $\backslash mathcal\text{'}(U)$ is finer than the subspace topology that this set inherits from $\backslash mathcal\text{'}(U)$).
The elements of $\backslash mathcal\text{'}(U)\; =\; (C^(U))\text{'}\_b$ can be identified as the space of distributions with compact support. Explicitly, if is a distribution on then the following are equivalent,
* $T\; \backslash in\; \backslash mathcal\text{'}(U)$;
* the support of is compact;
* the restriction of $T$ to $C\_c^(U),$ when that space is equipped with the subspace topology inherited from $C^(U)$ (a coarser topology than the canonical LF topology), is continuous;
* there is a compact subset of such that for every test function $\backslash phi$ whose support is completely outside of , we have $T(\backslash phi)=0.$
Compactly supported distributions define continuous linear functionals on the space $C^(U)$; recall that the topology on $C^(U)$ is defined such that a sequence of test functions $\backslash phi\_k$ converges to 0 if and only if all derivatives of $\backslash phi\_k$ 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 $C\_c^(U)$ to $C^(U).$

** Distributions of finite order **

Let $k\; \backslash in\; \backslash N.$ The natural inclusion $\backslash operatorname:\; C\_c^(U)\; \backslash to\; C\_c^k(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $^\backslash operatorname:\; (C\_c^k(U))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(U)\; =\; (C\_c^(U))\text{'}\_b$ is also a continuous injection. Consequently, the image of $^\backslash operatorname,$ denoted by $\backslash mathcal\text{'}^(U),$ forms a space of distributions when it is endowed with the strong dual topology of $(C\_c^k(U))\text{'}\_b$ (transferred to it via the transpose map $^\backslash operatorname:\; (C^(U))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}^(U),$ so $\backslash mathcal\text{'}^(U)$'s topology is finer than the subspace topology that this set inherits from $\backslash mathcal\text{'}(U)$). The elements of $\backslash mathcal\text{'}^(U)$ are ''the distributions of order ''. The distributions of order ≤ 0, which are also called ''distributions of order '', are exactly the distributions that are Radon measures (described above).
For $0\backslash neq\; k\; \backslash in\; \backslash N,$ a ''distribution of order '' is a distribution of order that is not a distribution of order .
A distribution is said to be of ''finite order'' if there is some integer such that it is a distribution of order , and the set of distributions of finite order is denoted by $\backslash mathcal\text{'}^(U).$ Note that if then $\backslash mathcal\text{'}^(U)\; \backslash subseteq\; \backslash mathcal\text{'}^(U)$ so that $\backslash mathcal\text{'}^(U)$ is a vector subspace of $\backslash mathcal\text{'}(U)$ and furthermore, if and only if $\backslash mathcal\text{'}^(U)\; =\; \backslash mathcal\text{'}(U).$
;Structure of distributions of finite order
Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is ''locally'' a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if $\backslash rho\_$ is the restriction mapping from to , then the image of $\backslash mathcal\text{'}(U)$ under $\backslash rho\_$ is contained in $\backslash mathcal\text{'}^(V).$
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:
:Theorem. Suppose $T\; \backslash in\; \backslash mathcal\text{'}(U)$ has finite order and $I\; =\backslash .$ Given any open subset of containing the support of , there is a family of Radon measures in , $(\backslash mu\_p)\_,$ such that for very $p\; \backslash in\; I,\; \backslash operatorname(\backslash mu\_p)\; \backslash subseteq\; V$ and
::$T\; =\; \backslash sum\_\; \backslash partial^p\; \backslash mu\_p.$
Example. (Distributions of infinite order) Let and for every test function $f,$ let
:$S\; f\; :=\; \backslash sum\_^\; (\backslash partial^\; f)\backslash left(\; \backslash frac\; \backslash right).$
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 ''tempered distributions'', which form a subspace of $\backslash mathcal\text{'}(\backslash R^n),$ the space of distributions on $\backslash R^n.$ This is a proper subspace: while every tempered distribution is a distribution and an element of $\backslash mathcal\text{'}(\backslash R^n),$ 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 $\backslash mathcal\text{'}(\backslash R^n).$
;Schwartz space
The Schwartz space, $\backslash mathcal(\backslash R^n),$ is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus $\backslash phi:\backslash R^n\backslash to\backslash R$ is in the Schwartz space provided that any derivative of $\backslash phi,$ 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 $\backslash alpha$ and $\backslash beta$ define:
:$p\_\; (\backslash phi)\; ~=~\; \backslash sup\_\; \backslash left\; |\; x^\backslash alpha\; \backslash partial^\backslash beta\; \backslash phi(x)\; \backslash right\; |.$
Then $\backslash phi$ is in the Schwartz space if all the values satisfy:
:$p\_\; (\backslash phi)\; <\; \backslash infty.$
The family of seminorms defines a locally convex topology on the Schwartz space. For ''n'' = 1, the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:
:$|f|\_\; =\; \backslash sup\_\; \backslash left(\backslash sup\_\; \backslash left\backslash \backslash right),\; \backslash qquad\; k,m\; \backslash in\; \backslash N.$
Otherwise, one can define a norm on $\backslash mathcal(\backslash R^n)$ via
:$\backslash |\backslash phi\; \backslash |\_\; ~=~\; \backslash max\_\; \backslash sup\_\; \backslash left|\; x^\backslash alpha\; \backslash partial^\backslash beta\; \backslash phi(x)\backslash ,\; \backslash qquad\; k\; \backslash ge\; 1.$
The Schwartz space is a Fréchet space (i.e. a complete metrizable locally convex space). Because the Fourier transform changes $\backslash partial^$ into multiplication by $x^$ and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence $\backslash $ in $\backslash mathcal(\backslash R^n)$ converges to 0 in $\backslash mathcal(\backslash R^n)$ if and only if the functions $(1\; +\; |x|)^k\; (\backslash partial^p\; f\_i)(x)$ converge to 0 uniformly in the whole of $\backslash R^n,$ which implies that such a sequence must converge to zero in $C^(\backslash R^n).$
$\backslash mathcal(\backslash R^n)$ is dense in $\backslash mathcal(\backslash R^n).$ The subset of all analytic Schwartz functions is dense in $\backslash mathcal(\backslash R^n)$ as well.
The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
:$\backslash mathcal(\backslash R^m)\; \backslash \; \backslash widehat\backslash \; \backslash mathcal(\backslash R^n)\; \backslash to\; \backslash mathcal(\backslash R^),$
where $\backslash widehat$ represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product).
;Tempered distributions
The natural inclusion $\backslash operatorname:\; \backslash mathcal(\backslash R^n)\; \backslash to\; \backslash mathcal(\backslash R^n)$ is a continuous injection whose image is dense in its codomain, so the transpose $^\backslash operatorname:\; (\backslash mathcal(\backslash R^n))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(\backslash R^n)$ is also a continuous injection. Thus, the image of the transpose map, denoted by $\backslash mathcal\text{'}(\backslash R^n),$ forms a space of distributions when it is endowed with the strong dual topology of $(\backslash mathcal(\backslash R^n))\text{'}\_b$ (transferred to it via the transpose map $^\backslash operatorname:\; (\backslash mathcal(\backslash R^n))\text{'}\_b\; \backslash to\; \backslash mathcal\text{'}(\backslash R^n),$ so the topology of $\backslash mathcal\text{'}(\backslash R^n)$ is finer than the subspace topology that this set inherits from $\backslash mathcal\text{'}(\backslash R^n)$).
The space $\backslash mathcal\text{'}(\backslash R^n)$ is called the space of ''>tempered distributions'' is it is the continuous dual of the Schwartz space. Equivalently, a distribution is a tempered distribution if and only if
:$\backslash left\; (\backslash text\; \backslash alpha,\; \backslash beta\; \backslash in\; \backslash N^n:\; \backslash lim\_\; p\_\; (\backslash phi\_m)\; =\; 0\; \backslash right\; )\; \backslash Longrightarrow\; \backslash lim\_\; T(\backslash phi\_m)=0.$
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) 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 $L^p(\backslash R^n)$ for are tempered distributions.
The ''tempered distributions'' can also be characterized as ''slowly growing'', meaning that each derivative of grows at most as fast as some polynomial. This characterization is dual to the ''rapidly falling'' behaviour of the derivatives of a function in the Schwartz space, where each derivative of $\backslash phi$ decays faster than every inverse power of . An example of a rapidly falling function is $|x|^n\backslash exp\; (-\backslash lambda\; |x|^\backslash beta)$ 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 $F\; :\; \backslash mathcal(\backslash R^n)\; \backslash to\; \backslash mathcal(\backslash R^n)$ is a TVS-automorphism of the Schwartz space, and ''the Fourier transform'' is defined to be its transpose $^F\; :\; \backslash mathcal\text{'}(\backslash R^n)\; \backslash to\; \backslash mathcal\text{'}(\backslash R^n),$ which (abusing notation) 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
:$F\; \backslash dfrac\; =\; ixFT$
and also with convolution: if is a tempered distribution and is a ''slowly increasing'' smooth function on $\backslash R^n,$ is again a tempered distribution and
:$F(\backslash psi\; T)\; =\; F\; \backslash psi\; *\; FT$
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 $T\; \backslash in\; \backslash mathcal\text{'}(\backslash R^n)$ is a tempered distribution, then there exists a constant , and positive integers and such that for all Schwartz functions $\backslash phi\; \backslash in\; \backslash mathcal(\backslash R^n)$
:$\backslash langle\; T,\; \backslash phi\; \backslash rangle\; \backslash le\; C\backslash sum\backslash nolimits\_\backslash sup\_\; \backslash left\; |x^\backslash alpha\; \backslash partial^\backslash beta\; \backslash phi(x)\; \backslash right\; |=C\backslash sum\backslash nolimits\_\; p\_(\backslash phi).$
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
:$T\; =\; \backslash partial^\backslash alpha\; F.$
Restriction of distributions to compact sets
If $T\; \backslash in\; \backslash mathcal\text{'}(\backslash R^n),$ then for any compact set $K\; \backslash subseteq\; \backslash R^n,$ there exists a continuous function compactly supported in $\backslash R^n$ (possibly on a larger set than itself) and a multi-index such that $T\; =\; \backslash partial^\backslash alpha\; F$ on $C\_c^(K).$

** 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.

** See also **

* Colombeau algebra
* Current (mathematics)
* Distribution (number theory)
* Distribution on a linear algebraic group
* Gelfand triple
* Generalized function
* Homogeneous distribution
* Hyperfunction
* Laplacian of the indicator
* Limit of a distribution
* Linear form
* Malgrange–Ehrenpreis theorem
* Pseudodifferential operator
* Riesz representation theorem
* Vague topology
* Weak solution

** Notes **

** References **

Bibliography

* * . * *. * . * . * . * . * * * * . * * * . * . * . * . * . * *

** Further reading **

*M. J. Lighthill (1959). ''Introduction to Fourier Analysis and Generalised Functions''. Cambridge University Press. (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
*V.S. Vladimirov (2002). ''Methods of the theory of generalized functions''. Taylor & Francis.
*.
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{{Hilbert space
Category:Generalized functions
Category:Functional analysis
Category:Smooth functions
Category:Generalizations of the derivative

Notation

The following notation will be used throughout this article: * $n$ is a fixed positive integer and $U$ is a fixed non-empty open subset of Euclidean space $\backslash R^.$ * $\backslash N\; =\; \backslash $ denotes the natural numbers. * $k$ will denote a non-negative integer or $\backslash infty.$ * If $f$ is a function then $\backslash operatorname(f)$ will denote its domain and the ''support of $f,$'' denoted by $\backslash operatorname(f),$ is defined to be the closure of the set $\backslash $ in $\backslash operatorname(f).$ * For two functions $f,\; g\; :\; U\; \backslash to\; \backslash Complex$, the following notation defines a canonical pairing: ::$\backslash langle\; f,\; g\backslash rangle\; :=\; \backslash int\_U\; f(x)\; g(x)\; \backslash ,dx.$ * A ''multi-index'' of size $n$ is an element in $\backslash N^n$ (given that $n$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be $n$). The ''length'' of a multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$ is defined as $\backslash alpha\_1+\backslash cdots+\backslash alpha\_n$ and denoted by $|\backslash alpha|.$ Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$: ::$\backslash begin\; x^\backslash alpha\; \&=\; x\_1^\; \backslash cdots\; x\_n^\; \backslash \backslash \; \backslash partial^\; \&=\; \backslash frac\; \backslash end$ :We also introduce a partial order of all multi-indices by $\backslash beta\; \backslash ge\; \backslash alpha$ if and only if $\backslash beta\_i\; \backslash ge\; \backslash alpha\_i$ for all $1\; \backslash le\; i\backslash le\; n.$ When $\backslash beta\; \backslash ge\; \backslash alpha$ we define their multi-index binomial coefficient as: ::$\backslash binom\; :=\; \backslash binom\; \backslash cdots\; \backslash binom.$ * $\backslash mathbb$ will denote a certain non-empty collection of compact subsets of $U$ (described in detail below).

= Basic properties

= ;Canonical LF topology's independence from One benefit of defining the canonical LF topology as the direct limit of a direct system is that we may immediately use the universal property of direct limits. Another benefit is that we can use well-known results from category theory to deduce that the canonical LF topology is actually independent of the particular choice of the directed collection of compact sets. And by considering different collections (in particular, those mentioned at the beginning of this article), we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes $C\_c^k(U)$ into a Hausdorff locally convex strict LF-space (and also a strict LB-space if ), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).If we take to be the set of ''all'' compact subsets of then we can use the universal property of direct limits to conclude that the inclusion $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C\_c^k(U)$ is a continuous and even that they are topological embedding for every compact subset . If however, we take to be the set of closures of some countable increasing sequence of relatively compact open subsets of having all of the properties mentioned earlier in this in this article then we immediately deduce that $C\_c^k(U)$ is a Hausdorff locally convex strict LF-space (and even a strict LB-space when ). All of these facts can also be proved directly without using direct systems (although with more work). ;Universal property From the universal property of direct limits, we know that if $u\; :\; C\_c^k(U)\; \backslash to\; Y$ is a linear map into a locally convex space (not necessarily Hausdorff), then is continuous if and only if is bounded if and only if for every , the restriction of to $C^k(K)$ is continuous (or bounded). ;Dependence of the canonical LF topology on Suppose is an open subset of $\backslash R^n$ containing $U.$ Let $I:\; C\_c^k(U)\backslash to\; C\_c^k(V)$ denote the map that sends a function in $C\_c^k(U)$ to its trivial extension on (which was defined above). This map is a continuous linear map. If (and only if) then $I(C\_c^(U))$ is ''not'' a dense subset of $C\_c^(V)$ and $I:\; C\_c^(U)\backslash to\; C\_c^(V)$ is ''not'' a topological embedding. Consequently, if then the transpose of $I:\; C\_c^(U)\backslash to\; C\_c^(V)$ is neither one-to-one nor onto. ;Bounded subsets A subset of $C\_c^k(U)$ is bounded in $C\_c^k(U)$ if and only if there exists some such that $B\; \backslash subseteq\; C^k(K)$ and is a bounded subset of $C^k(K).$ Moreover, if is compact and $S\; \backslash subseteq\; C^k(K)$ then is bounded in $C^k(K)$ if and only if it is bounded in $C^k(U).$ For any , any bounded subset of $C\_c^(U)$ (resp. $C^(U)$) is a relatively compact subset of $C\_c^k(U)$ (resp. $C^k(U)$), where . ;Non-metrizability For all compact , the interior of $C^k(K)$ in $C\_c^k(U)$ is empty so that $C\_c^k(U)$ is of the first category in itself. It follows from Baire's theorem that $C\_c^k(U)$ is ''not'' metrizable and thus also ''not'' normable (see this footnoteFor any TVS (metrizable or otherwise), the notion of completeness depends entirely on a certain so-called "canonical uniformity" that is defined using ''only'' the subtraction operation (see the article Complete topological vector space for more details). In this way, the notion of a complete TVS does not ''require'' the existence of any metric. However, if the TVS is metrizable and if is ''any'' translation-invariant metric on that defines its topology, then is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if the is a complete metric space. So if a TVS happens to have a topology that can be defined by such a metric then may be used to deduce the completeness of but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the Cartesian product of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS $\backslash R^\backslash N,$ which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on $\backslash R^\backslash N$). for an explanation of how the non-metrizable space $C\_c^k(U)$ can be complete even thought it does not admit a metric). The fact that $C\_c^(U)$ is a nuclear Montel space makes up for the non-metrizability of $C\_c^(U)$ (see this footnote for a more detailed explanation).One reason for giving $C\_c^(U)$ the canonical LF topology is because it is with this topology that $C\_c^(U)$ and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of topological vector spaces (TVSs) that are particularly similar to finite-dimensional Euclidean spaces: the Banach spaces (especially Hilbert spaces) and the nuclear Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are nuclear Montel Hilbert spaces but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. ''infinite'' differentiability, such as $C\_c^(U)$ and $C^(U)$) end up being nuclear TVSs while TVSs associated with ''finite'' continuous differentiability (such as $C^k(K)$ with compact and ) often end up being non-nuclear spaces, such as Banach spaces. ;Relationships between spaces Using the universal property of direct limits and the fact that the natural inclusions $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C^k(L)$ are all topological embedding, one may show that all of the maps $\backslash operatorname\_^\; :\; C^k(K)\; \backslash to\; C\_c^k(U)$ are also topological embeddings. Said differently, the topology on $C^k(K)$ is identical to the subspace topology that it inherits from $C\_c^k(U),$ where recall that $C^k(K)$'s topology was ''defined'' to be the subspace topology induced on it by $C^k(U).$ In particular, both $C\_c^k(U)$ and $C^k(U)$ induces the same subspace topology on $C^k(K).$ However, this does ''not'' imply that the canonical LF topology on $C\_c^k(U)$ is equal to the subspace topology induced on $C\_c^k(U)$ by $C^k(U)$; these two topologies on $C\_c^k(U)$ are in fact ''never'' equal to each other since the canonical LF topology is ''never'' metrizable while the subspace topology induced on it by $C^k(U)$ is metrizable (since recall that $C^k(U)$ is metrizable). The canonical LF topology on $C\_c^k(U)$ is actually ''strictly finer'' than the subspace topology that it inherits from $C^k(U)$ (thus the natural inclusion $C\_c^k(U)\backslash to\; C^k(U)$ is continuous but ''not'' a topological embedding). Indeed, the canonical LF topology is so fine that if $C\_c^(U)\backslash to\; X$ denotes some linear map that is a "natural inclusion" (such as $C\_c^(U)\backslash to\; C^k(U),$ or $C\_c^(U)\backslash to\; L^p(U),$ or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions, Radon measures, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on $C\_c^(U),$ the fine nature of the canonical LF topology means that more linear functionals on $C\_c^(U)$ end up being continuous ("more" means as compared to a coarser topology that we could have placed on $C\_c^(U)$ such as for instance, the subspace topology induced by some $C^k(U),$ which although it would have made $C\_c^(U)$ metrizable, it would have also resulted in fewer linear functionals on $C\_c^(U)$ being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making $C\_c^(U)$ into a complete TVS). ;Other properties * The differentiation map $C\_c^(U)\; \backslash to\; C\_c^(U)$ is a surjective continuous linear operator. * The bilinear multiplication map $C^(\backslash R^m)\; \backslash times\; C\_c^(\backslash R^n)\; \backslash to\; C\_c^(\backslash R^)$ given by $(f,g)\backslash mapsto\; fg$ is ''not'' continuous; it is however, hypocontinuous.

"The strong Pitkeev property for topological groups and topological vector spaces"

/ref> 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 ''not'' be defined using only sequences.

"Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"

(2017)T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36. which in particular implies that their topologies can ''not'' be defined entirely in terms of convergent sequences. A sequence $(f\_i)\_^$ in $C\_c^k(U)$ converges in $C\_c^k(U)$ if and only if there exists some such that $C^k(K)$ contains this sequence and this sequence converges in $C^k(K)$; equivalently, it converges if and only if the following two conditions hold: # There is a compact set containing the supports of all $f\_i.$ # For each multi-index , the sequence of partial derivatives $\backslash partial^\backslash alpha\; f\_$ tends uniformly to $\backslash partial^\backslash alpha\; f.$ Neither the space $C\_c^(U)$ nor its strong dual $\backslash mathcal\text{'}(U)$ is a sequential space, and consequently, their topologies can ''not'' be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is ''not'' enough to define the canonical LF topology on $C\_c^(U).$ The same can be said of the strong dual topology on $\backslash mathcal\text{'}(U).$ ;What sequences do characterize Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology, which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually ''define'' the convergence of a sequence of distributions; this is fine for sequences but it does ''not'' extend to the convergence of nets of distributions since a net may converge pointwise but fail to convergence in the strong dual topology). Sequences characterize continuity of linear maps valued in locally convex space. Suppose is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map into a locally convex space is continuous if and only if it maps null sequencesA ''null sequence'' is a sequence that converges to the origin. in to bounded subsets of .Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences. More generally, such a linear map is continuous if and only if it maps Mackey convergent null sequencesA sequence is said to be ''Mackey convergent to in $X,$'' if there exists a divergent sequence of positive real number such that is a bounded set in $X.$ to bounded subsets of $Y.$ So in particular, if a linear map into a locally convex space is sequentially continuous at the origin then it is continuous. However, this does ''not'' necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs. For every $k\; \backslash in\; \backslash ,\; C\_c^(U)$ is sequentially dense in $C\_c^k(U).$ Furthermore, $\backslash $ is a sequentially dense subset of $\backslash mathcal\text{'}(U)$ (with its strong dual topology) and also a sequentially dense subset of the strong dual space of $C^(U).$ ;Sequences of distributions A sequence of distributions $(T\_i)\_^$ converges with respect to the weak-* topology on $\backslash mathcal\text{'}(U)$ to a distribution if and only if :$\backslash langle\; T\_,\; f\; \backslash rangle\; \backslash to\; \backslash langle\; T,\; f\; \backslash rangle$ for every test function $f\; \backslash in\; \backslash mathcal(U).$ For example, if $f\_m:\backslash R\backslash to\backslash R$ is the function :$f\_m(x)\; =\; \backslash begin\; m\; \&\; \backslash text\; x\; \backslash in,\backslash frac\backslash \backslash \; 0\; \&\; \backslash text\; \backslash end$ and is the distribution corresponding to $f\_m,$ then :$\backslash langle\; T\_m,\; f\; \backslash rangle\; =\; m\; \backslash int\_0^\; f(x)\backslash ,\; dx\; \backslash to\; f(0)\; =\; \backslash langle\; \backslash delta,\; f\; \backslash rangle$ as , so in $\backslash mathcal\text{'}(\backslash R).$ Thus, for large , the function $f\_m$ can be regarded as an approximation of the Dirac delta distribution. ;Other properties * The strong dual space of $\backslash mathcal\text{'}(U)$ is TVS isomorphic to $C\_c^(U)$ via the canonical TVS-isomorphism $C\_c^(U)\; \backslash to\; (\backslash mathcal\text{'}(U))\text{'}\_$ defined by sending $f\; \backslash in\; C\_c^(U)$ to ''value at $f$'' (i.e. to the linear functional on $\backslash mathcal\text{'}(U)$ defined by sending $d\; \backslash in\; \backslash mathcal\text{'}(U)$ to $d(f)$); * On any bounded subset of $\backslash mathcal\text{'}(U),$ the weak and strong subspace topologies coincide; the same is true for $C\_c^(U)$; * Every weakly convergent sequence in $\backslash mathcal\text{'}(U)$ is strongly convergent (although this does not extend to nets).

= Problem of multiplying distributions

= It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if p.v. is the distribution obtained by the Cauchy principal value :$\backslash left(\backslash operatorname\backslash frac\backslash right)(\backslash phi)\; =\; \backslash lim\_\; \backslash int\_\; \backslash frac\backslash ,\; dx\; \backslash quad\; \backslash text\; \backslash phi\; \backslash in\; \backslash mathcal(\backslash R).$ If is the Dirac delta distribution then :$(\backslash delta\; \backslash times\; x)\; \backslash times\; \backslash operatorname\; \backslash frac\; =\; 0$ but :$\backslash delta\; \backslash times\; \backslash left(x\; \backslash times\; \backslash operatorname\; \backslash frac\backslash right)\; =\; \backslash delta$ 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. 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.

Bibliography

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