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{{Short description|Mathematical analysis term similar to generalized function}}
{{about|generalized functions in mathematical analysis|the concept of distributions in probability theory|Probability distribution|artificial landscapes|Test functions for optimization|other uses|Distribution (disambiguation)#In mathematics{{!}}Distribution § Mathematics}}

'''Distributions''', also known as '''Schwartz distributions''' or '''[[generalized function]]s''', are objects that generalize the classical notion of functions in [[mathematical analysis]]. Distributions make it possible to [[derivative|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 equation]]s, where it may be easier to establish the existence of distributional solutions ([[weak solution]]s) than [[Solution of a differential equation|classical solutions]], or where appropriate classical solutions may not exist. Distributions are also important in [[physics]] and [[engineering]] where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the [[Dirac delta function|Dirac delta]] function.

A [[Function (mathematics)|function]] <math>f</math> is normally thought of as {{em|[[Group action|acting]]}} on the {{em|points}} in the function [[Domain (function)|domain]] by "sending" a point <math>x</math> in the domain to the point <math>f(x).</math> Instead of acting on points, distribution theory reinterprets functions such as <math>f</math> as acting on {{em|test functions}} in a certain way. In applications to physics and engineering, '''{{em|test functions}}''' are usually [[Smooth function|infinitely differentiable]] [[Complex number|complex]]-valued (or [[Real number|real]]-valued) functions with [[Compact space|compact]] [[Support (mathematics)|support]] that are defined on some given non-empty [[Open set|open subset]] <math>U \subseteq \R^n</math>. ([[Bump function]]s are examples of test functions.) The set of all such test functions forms a [[vector space]] that is denoted by <math>C_c^\infty(U)</math> or <math>\mathcal{D}(U).</math>

Most commonly encountered functions, including all [[Continuous function|continuous]] maps <math>f : \R \to \R</math> if using <math>U := \R,</math> can be canonically reinterpreted as acting via "[[Integral|integration]] against a test function." Explicitly, this means that such a function <math>f</math> "acts on" a test function <math>\psi \in \mathcal{D}(\R)</math> by "sending" it to the [[Integral (mathematics)|number]] <math display=inline>\int_\R f \, \psi \, dx,</math> which is often denoted by <math>D_f(\psi).</math> This new action <math display=inline>\psi \mapsto D_f(\psi)</math> of <math>f</math> defines a [[Functional (mathematics)|scalar-valued map]] <math>D_f : \mathcal{D}(\R) \to \Complex,</math> whose domain is the space of test functions <math>\mathcal{D}(\R).</math> This [[Functional (mathematics)|functional]] <math>D_f</math> turns out to have the two defining properties of what is known as a {{em|distribution on <math>U = \R</math>}}: it is [[Linear form|linear]], and it is also [[Continuous function|continuous]] when <math>\mathcal{D}(\R)</math> is given a certain [[topology]] called {{em|the canonical LF topology}}. The action (the integration <math display=inline>\psi \mapsto \int_\R f \, \psi \, dx</math>) of this distribution <math>D_f</math> on a test function <math>\psi</math> can be interpreted as a weighted average of the distribution on the [[Support (mathematics)|support]] of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like <math>D_f</math> that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the [[Dirac delta function]] and distributions defined to act by integration of test functions <math display=inline>\psi \mapsto \int_U \psi d \mu</math> against certain [[Measure (mathematics)|measures]] <math>\mu</math> on <math>U.</math> Nonetheless, it is still always possible to [[#Decomposition of distributions as sums of derivatives of continuous functions|reduce any arbitrary distribution]] down to a simpler {{em|family}} of related distributions that do arise via such actions of integration.

More generally, a {{em|'''distribution''' on <math>U</math>}} is by definition a [[Linear form|linear functional]] on <math>C_c^\infty(U)</math> that is [[Continuous linear functional|continuous]] when <math>C_c^\infty(U)</math> is given a topology called the '''{{em|canonical LF topology}}'''. This leads to {{em|the}} space of (all) distributions on <math>U</math>, usually denoted by <math>\mathcal{D}'(U)</math> (note the [[Prime (symbol)|prime]]), which by definition is the [[Vector space|space]] of all distributions on <math>U</math> (that is, it is the [[continuous dual space]] of <math>C_c^\infty(U)</math>); it is these distributions that are the main focus of this article.

Definitions of the appropriate topologies on [[spaces of test functions and distributions]] are given in the article on [[spaces of test functions and distributions]]. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.

{{TOCLimit}}

==History==

The practical use of distributions can be traced back to the use of [[Green's function|Green's functions]] in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to {{harvtxt|Kolmogorov|Fomin|1957}}, generalized functions originated in the work of {{harvs|txt|author-link=Sergei Lvovich Sobolev|first=Sergei|last= Sobolev|year=1936}} on [[second-order differential equation|second-order]] [[hyperbolic partial differential equation]]s, 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. {{harvtxt|Gårding|1997}} comments that although the ideas in the transformative book by {{harvtxt|Schwartz|1951}} were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference. A detailed history of the theory of distributions was given by {{harvtxt|Lützen|1982}}.

==Notation==

The following notation will be used throughout this article:

* <math>n</math> is a fixed positive integer and <math>U</math> is a fixed non-empty [[Open set|open subset]] of [[Euclidean space]] <math>\R^n.</math>
* <math>\N_0 = \{0, 1, 2, \ldots\}</math> denotes the [[natural number]]s.
* <math>k</math> will denote a non-negative integer or <math>\infty.</math>
* If <math>f</math> is a [[Function (mathematics)|function]] then <math>\operatorname{Dom}(f)</math> will denote its [[Domain of a function|domain]] and the '''{{em|[[Support (mathematics)|{{visible anchor|support of a function|text=support}}]]}}''' of <math>f,</math> denoted by <math>\operatorname{supp}(f),</math> is defined to be the [[Closure (topology)|closure]] of the set <math>\{x \in \operatorname{Dom}(f): f(x) \neq 0\}</math> in <math>\operatorname{Dom}(f).</math>
* For two functions <math>f, g : U \to \Complex,</math> the following notation defines a canonical [[Dual system|pairing]]: <math display=block>\langle f, g\rangle := \int_U f(x) g(x) \,dx.</math>
* A '''{{em|[[Multi-index notation|multi-index]]}} of size''' <math>n</math> is an element in <math>\N^n</math> (given that <math>n</math> is fixed, if the size of multi-indices is omitted then the size should be assumed to be <math>n</math>). The '''{{em|length}}''' of a multi-index <math>\alpha = (\alpha_1, \ldots, \alpha_n) \in \N^n</math> is defined as <math>\alpha_1+\cdots+\alpha_n</math> and denoted by <math>|\alpha|.</math> Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index <math>\alpha = (\alpha_1, \ldots, \alpha_n) \in \N^n</math>: <math display=block>\begin{align}
x^\alpha &= x_1^{\alpha_1} \cdots x_n^{\alpha_n} \\
\partial^\alpha &= \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}
\end{align}</math> We also introduce a partial order of all multi-indices by <math>\beta \ge \alpha</math> if and only if <math>\beta_i \ge \alpha_i</math> for all <math>1 \le i\le n.</math> When <math>\beta \ge \alpha</math> we define their multi-index binomial coefficient as: <math display=block>\binom{\beta}{\alpha} := \binom{\beta_1}{\alpha_1} \cdots \binom{\beta_n}{\alpha_n}.</math>
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==Basic idea==
[[File:Mollifier Illustration.svg|right|thumb|280px|A typical test function, the [[bump function]] <math>\Psi(x).</math> It is [[smooth function|smooth]] (infinitely differentiable) and has [[compact support]] (is zero outside an interval, in this case the interval <math>[-1, 1]</math>).]]

Distributions are a class of [[linear functional]]s that map a set of {{em|test functions}} (conventional and [[well-behaved]] functions) into the set of real or complex numbers. In the simplest case, the set of test functions considered is <math>\mathcal{D}(\R),</math> which is the set of functions <math>\varphi:\R\to\R</math> having two properties:

* <math>\varphi</math> is [[Smooth function|smooth]] (infinitely differentiable);
* <math>\varphi</math> has [[compact support]] (is identically zero outside some bounded interval).

A distribution {{mvar|T}} is a continuous linear mapping <math>T:\mathcal{D}(\R)\to\R.</math> Instead of writing <math>T(\varphi),</math> it is conventional to write <math>\langle T, \varphi \rangle</math> for the value of <math>T</math> acting on a test function <math>\varphi.</math> A simple example of a distribution is the [[Dirac delta]], <math>\delta,</math> defined by
<math display=block>\langle \delta, \varphi \rangle = \varphi(0),</math>
meaning that <math>\delta</math> evaluates a test function at {{math|0}}. Its physical interpretation is as the density of a point source.

As described next, there are straightforward mappings from both [[locally integrable function]]s and [[Radon measure]]s to corresponding distributions, but not all distributions can be formed in this manner.

===Functions and measures as distributions===
Suppose <math>f : \R \to \R</math> is a [[locally integrable function]]. Then a corresponding distribution, denoted by <math>T_f,</math> may be defined by
<math display=block>\langle T_f, \varphi \rangle = \int_\R f(x) \varphi(x) \,dx \qquad \text{for } \varphi \in \mathcal{D}(\R).</math>

This integral is a [[real number]] which depends [[Linear operator|linearly]] and [[Continuous function|continuously]] on <math>\varphi.</math> Conversely, the values of the distribution <math>T_f</math> on test functions in <math>\mathcal{D}(\R)</math> determine the pointwise almost everywhere values of the function <math>f</math> on <math>\R.</math> In a conventional [[abuse of notation]], <math>f</math> is often used to represent both the original function <math>f</math> and the corresponding distribution <math>T_f..</math> This example suggests the definition of a distribution as a linear and, in an appropriate sense, continuous [[Functional (mathematics)|functional]] on the space of test functions <math>\mathcal{D}(\R).</math>

Similarly, if <math>\mu</math> is a [[Radon measure]] on <math>\R,</math> then a corresponding distribution, denoted by <math>R_{\mu},</math> may be defined by
<math display=block>\left\langle R_\mu, \varphi \right\rangle = \int_\R \varphi\, d\mu \qquad \text{ for } \varphi \in \mathcal{D}(\R).</math>

This integral also depends linearly and continuously on <math>\varphi,</math> so that <math>R_{\mu}</math> is a distribution. If <math>\mu</math> is [[absolute continuity|absolutely continuous]] with respect to Lebesgue measure with density <math>f</math> and <math>d \mu = f\,dx,</math> then this definition for <math>R_{\mu}</math> is the same as the previous one for <math>T_f,</math> but if <math>\mu</math> is not absolutely continuous, then <math>R_{\mu}</math> is a distribution that is not associated with a function. For example, if <math>P</math> is the point-mass measure on <math>\R</math> that assigns measure one to the singleton set <math>\{0\}</math> and measure zero to sets that do not contain zero, then
<math display=block>\int_\R \varphi\, dP = \varphi(0),</math>
so that <math>R_P = \delta</math> is the Dirac delta.

===Adding and multiplying distributions===
Distributions may be multiplied by real numbers and added together, so they form a real [[vector space]]. A distribution may also be multiplied by a rapidly decreasing infinitely differentiable function to get another distribution, but [[Distribution (mathematics)#Problem of multiplication|it is not possible to define a product of general distributions]] that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by {{harvtxt|Schwartz|1954}}, and is usually referred to as the {{em|Schwartz Impossibility Theorem}}.

===Derivatives of distributions===
It is desirable to choose a definition for the derivative of a distribution which, at least for distributions derived from smooth functions, has the property that <math>T'_f = T_{f'}</math> (i.e. <math>(T_f)' = T_{(f')}</math> where <math>f'</math> is the usual derivative of <math>f</math> and <math>(T_f)'</math> denotes the derivative of the distribution <math>T_f,</math> which we wish to define). If <math>\phi</math> is a test function, we can use [[integration by parts]] to see that
<math display=block>\langle f', \phi \rangle = \int_\R f'\phi \,dx = \Big[ f(x) \phi(x) \Big]_{-\infty}^\infty -\int_\R f \phi' \,dx = -\langle f, \phi' \rangle</math>
where the last equality follows from the fact that <math>\phi</math> has compact support, so is zero outside of a bounded set.
This suggests that if {{mvar|T}} is a {{em|distribution}}, we should define its derivative <math>T'</math> by
<math display=block>\langle T', \phi \rangle = - \langle T, \phi' \rangle.</math>

It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.

'''Example''': Recall that the [[Dirac delta]] (i.e. the so-called Dirac delta "function") is the distribution defined by the equation
<math display=block>\langle \delta, \phi \rangle = \phi(0).</math>

It is the derivative of the distribution corresponding to the [[Heaviside step function]] <math>H</math>: For any test function <math>\phi</math>
<math display=block>\langle H', \phi \rangle = -\int_{-\infty}^\infty H(x) \phi'(x) \, dx = -\phi(\infty) + \phi(0) = \langle \delta, \phi \rangle,</math>
so <math>H = \delta.</math> Note, <math>\phi(\infty)=0</math> because <math>\phi</math> has compact support by our definition of a test function. Similarly, the derivative of the Dirac delta is the distribution defined by the equation
<math display=block>\langle \delta', \phi \rangle = - \phi'(0).</math>

This latter distribution is an example of a distribution that is not derived from a function or a measure. Its physical interpretation is the density of a dipole source. Just as the Dirac impulse can be realized in the weak limit as a sequence of various kinds of constant norm bump functions of ever increasing amplitude and narrowing support, its derivative can by definition be realized as the weak limit of the negative derivatives of said functions, which are now antisymmetric about the eventual distribution's point of singular support.-->

==Definitions of test functions and distributions==

In this section, some basic notions and definitions needed to define real-valued distributions on {{mvar|U}} are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on [[spaces of test functions and distributions]].

{{block indent|em=1.5|text='''Notation''': 
# Let <math>k \in \{0, 1, 2, \ldots, \infty\}.</math>
# Let <math>C^k(U)</math> denote the [[vector space]] of all {{mvar|k}}-times [[continuously differentiable]] real or complex-valued functions on {{mvar|U}}.
# For any compact subset <math>K \subseteq U,</math> let <math>C^k(K)</math> and <math>C^k(K;U)</math> both denote the vector space of all those functions <math>f \in C^k(U)</math> such that <math>\operatorname{supp}(f) \subseteq K.</math>
#* If <math>f \in C^k(K)</math> then the domain of <math>f</math> is {{mvar|U}} and not {{mvar|K}}. So although <math>C^k(K)</math> depends on both {{mvar|K}} and {{mvar|U}}, only {{mvar|K}} is typically indicated. The justification for this common practice is [[#Omitting the open set from notation|detailed below]]. The notation <math>C^k(K;U)</math> will only be used when the notation <math>C^k(K)</math> risks being ambiguous.
#* Every <math>C^k(K)</math> contains the constant {{math|0}} map, even if <math>K = \varnothing.</math>
# Let <math>C_c^k(U)</math> denote the set of all <math>f \in C^k(U)</math> such that <math>f \in C^k(K)</math> for some compact subset {{mvar|K}} of {{mvar|U}}.
#* Equivalently, <math>C_c^k(U)</math> is the set of all <math>f \in C^k(U)</math> such that <math>f</math> has compact [[#support of a function|support]].
#* <math>C_c^k(U)</math> is equal to the union of all <math>C^k(K)</math> as <math>K \subseteq U</math> ranges over all compact subsets of <math>U.</math>
#* If <math>f</math> is a real-valued function on <math>U</math>, then <math>f</math> is an element of <math>C_c^k(U)</math> if and only if <math>f</math> is a <math>C^k</math> [[bump function]]. Every real-valued test function on <math>U</math> is also a complex-valued test function on <math>U.</math>
}}

[[File:Bump.png|thumb|350x350px|The graph of the [[bump function]] <math>(x,y) \in \R^2 \mapsto \Psi(r),</math> where <math>r = \left(x^2 + y^2\right)^\frac{1}{2}</math> and <math>\Psi(r) = e^{-\frac{1}{1 - r^2}}\cdot\mathbf{1}_{\{|r|<1\}}.</math> This function is a test function on <math>\R^2</math> and is an element of <math>C^\infty_c\left(\R^2\right).</math> The [[#support of a function|support]] of this function is the closed [[unit disk]] in <math>\R^2.</math> It is non-zero on the open unit disk and it is equal to {{math|0}} everywhere outside of it.]]

For all <math>j, k \in \{0, 1, 2, \ldots, \infty\}</math> and any compact subsets <math>K</math> and <math>L</math> of <math>U</math>, we have:
<math display=block>\begin{align}
C^k(K) &\subseteq C^k_c(U) \subseteq C^k(U) \\
C^k(K) &\subseteq C^k(L) && \text{if } K \subseteq L \\
C^k(K) &\subseteq C^j(K) && \text{if } j \le k \\
C_c^k(U) &\subseteq C^j_c(U) && \text{if } j \le k \\
C^k(U) &\subseteq C^j(U) && \text{if } j \le k \\
\end{align}</math>

{{block indent|em=1.5|text='''Definition''': Elements of <math>C_c^\infty(U)</math> are called '''{{em|test functions}}''' on {{mvar|U}} and <math>C_c^\infty(U)</math> is called the '''{{em|space of test functions}}''' on {{mvar|U}}. We will use both <math>\mathcal{D}(U)</math> and <math>C_c^\infty(U)</math> to denote this space.}}

Distributions on {{mvar|U}} are [[continuous linear functional]]s on <math>C_c^\infty(U)</math> when this vector space is endowed with a particular topology called the '''{{em|canonical LF-topology}}'''. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on <math>C_c^\infty(U)</math> that are often straightforward to verify.

'''Proposition''': A [[Linear form|linear functional]] {{mvar|T}} on <math>C_c^\infty(U)</math> is continuous, and therefore a '''{{em|distribution}}''', if and only if any of the following equivalent conditions is satisfied:

# For every compact subset <math>K\subseteq U</math> there exist constants <math>C>0</math> and <math>N\in \N</math> (dependent on <math>K</math>) such that for all <math>f \in C_c^\infty(U)</math> with [[#support of a function|support]] contained in <math>K</math>,{{sfn|Trèves|2006|pp=222-223}}<ref>{{harvnb|Grubb|2009|page=14}}</ref> <math display="block">|T(f)| \leq C \sup \{|\partial^\alpha f(x)|: x \in U, |\alpha| \leq N\}.</math>
# For every compact subset <math>K\subseteq U</math> and every sequence <math>\{f_i\}_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> whose supports are contained in <math>K</math>, if <math>\{\partial^\alpha f_i\}_{i=1}^\infty</math> converges uniformly to zero on <math>U</math> for every [[multi-index]] <math>\alpha</math>, then <math>T(f_i) \to 0.</math>

===Topology on ''C''<sup>''k''</sup>(''U'')===

We now introduce the [[seminorm]]s that will define the topology on <math>C^k(U).</math> 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.

{{block indent|em=1.5|text=Suppose <math>k \in \{0, 1, 2, \ldots, \infty\}</math> and <math>K</math> is an arbitrary compact subset of <math>U.</math> Suppose <math>i</math> is an integer such that <math>0 \leq i \leq k</math><ref group=note>Note that <math>i</math> being an integer implies <math>i \neq \infty.</math> This is sometimes expressed as <math>0 \leq i < k + 1.</math> Since <math>\infty + 1 = \infty,</math> the inequality "<math>0 \leq i < k + 1</math>" means: <math>0 \leq i < \infty</math> if <math>k = \infty,</math> while if <math>k \neq \infty</math> then it means <math>0 \leq i \leq k.</math></ref> and <math>p</math> is a multi-index with length <math>| p|\leq k.</math> For <math>K \neq \varnothing</math> and <math>f \in C^k(U),</math> define:
<math display=block>\begin{alignat}{4}
\text{ (1) }\ & s_{p,K}(f) &&:= \sup_{x_0 \in K} \left| \partial^p f(x_0) \right| \\[4pt]
\text{ (2) }\ & q_{i,K}(f) &&:= \sup_{|p| \leq i} \left(\sup_{x_0 \in K} \left| \partial^p f(x_0) \right|\right) = \sup_{|p| \leq i} \left(s_{p, K}(f)\right) \\[4pt]
\text{ (3) }\ & r_{i,K}(f) &&:= \sup_{\stackrel{|p| \leq i}{x_0 \in K}}  \left| \partial^p f(x_0) \right| \\[4pt]
\text{ (4) }\ & t_{i,K}(f) &&:= \sup_{x_0 \in K} \left(\sum_{|p| \leq i} \left| \partial^p f(x_0) \right|\right)
\end{alignat}</math>
while for <math>K = \varnothing,</math> define all the functions above to be the constant {{math|0}} map. 
}}
All of the functions above are non-negative <math>\R</math>-valued<ref group="note">The image of the [[compact set]] <math>K</math> under a continuous <math>\R</math>-valued map (for example, <math>x \mapsto \left|\partial^p f(x)\right|</math> for <math>x \in U</math>) is itself a compact, and thus bounded, subset of <math>\R.</math> If <math>K \neq \varnothing</math> then this implies that each of the functions defined above is <math>\R</math>-valued (that is, none of the [[Infimum and supremum|supremums]] above are ever equal to <math>\infty</math>).</ref> [[seminorm]]s on <math>C^k(U).</math> As explained in [[Locally convex topological vector space#Definition via seminorms|this article]], every set of seminorms on a vector space induces a [[Locally convex topological vector space|locally convex]] [[Topological vector space|vector topology]].

Each of the following sets of seminorms 
<math display=block>\begin{alignat}{4}
A ~:= \quad &\{q_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\
B ~:= \quad &\{r_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\
C ~:= \quad &\{t_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\
D ~:= \quad &\{s_{p,K} &&: \;K \text{ compact and } \;&&p \in \N^n \text{ satisfies } \;&&|p| \leq k\}
\end{alignat}</math>
generate the same [[Locally convex topological vector space|locally convex]] [[Topological vector space|vector topology]] on <math>C^k(U)</math> (so for example, the topology generated by the seminorms in <math>A</math> is equal to the topology generated by those in <math>C</math>). 

{{block indent|em=1.5|text=The vector space <math>C^k(U)</math> is endowed with the [[Locally convex topological vector space|locally convex]] topology induced by any one of the four families <math>A, B, C, D</math> of seminorms described above. This topology is also equal to the vector topology induced by {{em|all}} of the seminorms in <math>A \cup B \cup C \cup D.</math>}}

With this topology, <math>C^k(U)</math> becomes a locally convex [[Fréchet space]] that is {{em|not}} [[Normable space|normable]]. Every element of <math>A \cup B \cup C \cup D</math> is a continuous seminorm on <math>C^k(U).</math>
Under this topology, a [[net (mathematics)|net]] <math>(f_i)_{i\in I}</math> in <math>C^k(U)</math> converges to <math>f \in C^k(U)</math> if and only if for every multi-index <math>p</math> with <math>|p|< k + 1</math> and every compact <math>K,</math> the net of partial derivatives <math>\left(\partial^p f_i\right)_{i \in I}</math> [[Uniform convergence|converges uniformly]] to <math>\partial^p f</math> on <math>K.</math>{{sfn|Trèves|2006|pp=85-89}} For any <math>k \in \{0, 1, 2, \ldots, \infty\},</math> any [[Bounded set (topological vector space)|(von Neumann) bounded subset]] of <math>C^{k+1}(U)</math> is a [[relatively compact]] subset of <math>C^k(U).</math>{{sfn|Trèves|2006|pp=142-149}} In particular, a subset of <math>C^\infty(U)</math> is bounded if and only if it is bounded in <math>C^i(U)</math> for all <math>i \in \N.</math>{{sfn|Trèves|2006| pp=142-149}} The space <math>C^k(U)</math> is a [[Montel space]] if and only if <math>k = \infty.</math>{{sfn|Trèves|2006|pp=356-358}}

A subset <math>W</math> of <math>C^\infty(U)</math> is open in this topology if and only if there exists <math>i\in \N</math> such that <math>W</math> is open when <math>C^\infty(U)</math> is endowed with the [[subspace topology]] induced on it by <math>C^i(U).</math>

====Topology on ''C''<sup>''k''</sup>(''K'')====

As before, fix <math>k \in \{0, 1, 2, \ldots, \infty\}.</math> Recall that if <math>K</math> is any compact subset of <math>U</math> then <math>C^k(K) \subseteq C^k(U).</math>

{{block indent|em=1.5|text='''Assumption''': For any compact subset <math>K \subseteq U,</math> we will henceforth assume that <math>C^k(K)</math> is endowed with the [[subspace topology]] it inherits from the [[Fréchet space]] <math>C^k(U).</math>}}

If <math>k</math> is finite then <math>C^k(K)</math> is a [[Banach space]]{{sfn|Trèves|2006|pp=131-134}} with a topology that can be defined by the [[Norm (mathematics)|norm]]
<math display=block>r_K(f) := \sup_{|p|<k} \left( \sup_{x_0 \in K} \left|\partial^p f(x_0)\right| \right).</math>
And when <math>k = 2,</math> then <math>C^k(K)</math> is even a [[Hilbert space]].{{sfn|Trèves|2006|pp=131-134}}

====Trivial extensions and independence of ''C''<sup>''k''</sup>(''K'')'s topology from ''U''====
{{anchor|Omitting the open set from notation}}

Suppose <math>U</math> is an open subset of <math>\R^n</math> and <math>K \subseteq U</math> is a compact subset. By definition, elements of <math>C^k(K)</math> are functions with domain <math>U</math> (in symbols, <math>C^k(K) \subseteq C^k(U)</math>), so the space <math>C^k(K)</math> and its topology depend on <math>U;</math> to make this dependence on the open set <math>U</math> clear, temporarily denote <math>C^k(K)</math> by <math>C^k(K;U).</math> 
Importantly, changing the set <math>U</math> to a different open subset <math>U'</math> (with <math>K \subseteq U'</math>) will change the set <math>C^k(K)</math> from <math>C^k(K;U)</math> to <math>C^k(K;U'),</math><ref group="note">Exactly as with <math>C^k(K;U),</math> the space <math>C^k(K; U')</math> is defined to be the vector subspace of <math>C^k(U')</math> consisting of maps with [[#support of a function|support]] contained in <math>K</math> endowed with the subspace topology it inherits from <math>C^k(U')</math>.</ref> so that elements of <math>C^k(K)</math> will be functions with domain <math>U'</math> instead of <math>U.</math> 
Despite <math>C^k(K)</math> depending on the open set (<math>U \text{ or } U'</math>), the standard notation for <math>C^k(K)</math> makes no mention of it. 
This is justified because, as this subsection will now explain, the space <math>C^k(K;U)</math> is canonically identified as a subspace of <math>C^k(K;U')</math> (both algebraically and topologically). 

It is enough to explain how to canonically identify <math>C^k(K; U)</math> with <math>C^k(K; U')</math> when one of <math>U</math> and <math>U'</math> is a subset of the other. The reason is that if <math>V</math> and <math>W</math> are arbitrary open subsets of <math>\R^n</math> containing <math>K</math> then the open set <math>U := V \cap W</math> also contains <math>K,</math> so that each of <math>C^k(K; V)</math> and <math>C^k(K; W)</math> is canonically identified with <math>C^k(K; V \cap W)</math> and now by transitivity, <math>C^k(K; V)</math> is thus identified with <math>C^k(K; W).</math> 
So assume <math>U \subseteq V</math> are open subsets of <math>\R^n</math> containing <math>K.</math>

Given <math>f \in C_c^k(U),</math> its {{em|'''trivial extension''' to <math>V</math>}} is the function <math>F : V \to \Complex</math> defined by:
<math display=block>F(x) = \begin{cases}
f(x) & x \in U, \\
0 & \text{otherwise}.
\end{cases}</math>
This trivial extension belongs to <math>C^k(V)</math> (because <math>f \in C_c^k(U)</math> has compact support) and it will be denoted by <math>I(f)</math> (that is, <math>I(f) := F</math>). The assignment <math>f \mapsto I(f)</math> thus induces a map <math>I : C_c^k(U) \to C^k(V)</math> that sends a function in <math>C_c^k(U)</math> to its trivial extension on <math>V.</math> This map is a linear [[Injective function|injection]] and for every compact subset <math>K \subseteq U</math> (where <math>K</math> is also a compact subset of <math>V</math> since <math>K \subseteq U \subseteq V</math>), 
<math display=block>\begin{alignat}{4}
I\left(C^k(K; U)\right) &~=~ C^k(K; V) \qquad \text{ and thus } \\
I\left(C_c^k(U)\right) &~\subseteq~ C_c^k(V).
\end{alignat}</math>
If <math>I</math> is restricted to <math>C^k(K; U)</math> then the following induced linear map is a [[homeomorphism]] (linear homeomorphisms are called {{em|[[TVS-isomorphism]]s}}):
<math display=block>\begin{alignat}{4}
 \,& C^k(K; U) && \to \,&& C^k(K;V) \\
   & f                  && \mapsto\,&& I(f) \\
\end{alignat}</math>
and thus the next map is a [[topological embedding]]:
<math display=block>\begin{alignat}{4}
 \,& C^k(K; U) && \to \,&& C^k(V) \\
   & f                  && \mapsto\,&& I(f). \\
\end{alignat}</math>
Using the injection
<math display=block>I : C_c^k(U) \to C^k(V)</math>
the vector space <math>C_c^k(U)</math> is canonically identified with its image in <math>C_c^k(V) \subseteq C^k(V).</math> Because <math>C^k(K; U) \subseteq C_c^k(U),</math> through this identification, <math>C^k(K; U)</math> can also be considered as a subset of <math>C^k(V).</math>
Thus the topology on <math>C^k(K;U)</math> is independent of the open subset <math>U</math> of <math>\R^n</math> that contains <math>K,</math>{{sfn|Rudin|1991|pp=149-181}} which justifies the practice of writing <math>C^k(K)</math> instead of <math>C^k(K; U).</math>

===Canonical LF topology===
{{Main|Spaces of test functions and distributions}}
{{See also|LF-space|Topology of uniform convergence}}

Recall that <math>C_c^k(U)</math> denotes all functions in <math>C^k(U)</math> that have compact [[#support of a function|support]] in <math>U,</math> where note that <math>C_c^k(U)</math> is the union of all <math>C^k(K)</math> as <math>K</math> ranges over all compact subsets of <math>U.</math> Moreover, for each <math>k,\, C_c^k(U)</math> is a dense subset of <math>C^k(U).</math> The special case when <math>k = \infty</math> gives us the space of test functions.

{{block indent|em=1.5|text=<math>C_c^\infty(U)</math> is called the {{em|'''space of test functions''' on <math>U</math>}} and it may also be denoted by <math>\mathcal{D}(U).</math> Unless indicated otherwise, it is endowed with a topology called '''{{em|the canonical LF topology}}''', whose definition is given in the article: [[Spaces of test functions and distributions]].}}

The canonical LF-topology is {{em|not}} metrizable and importantly, it is [[Comparison of topologies|{{em|'''strictly''' finer}}]] than the [[subspace topology]] that <math>C^\infty(U)</math> induces on <math>C_c^\infty(U).</math> However, the canonical LF-topology does make <math>C_c^\infty(U)</math> into a [[Complete topological vector space|complete]] [[Reflexive space|reflexive]] [[Nuclear space|nuclear]]{{sfn|Trèves|2006|pp=526-534}} [[Montel space|Montel]]{{sfn|Trèves|2006|p=357}} [[Bornological space|bornological]] [[Barrelled space|barrelled]] [[Mackey space]]; the same is true of its [[strong dual space]] (that is, the space of all distributions with its usual topology). The canonical [[LF-space|LF-topology]] can be defined in various ways.

===Distributions===

{{See also|Continuous linear functional}}

As discussed earlier, continuous [[Linear form|linear functionals]] on a <math>C_c^\infty(U)</math> are known as distributions on <math>U.</math> Other equivalent definitions are described below.

{{block indent|em=1.5|text=By definition, a {{em|'''distribution''' on <math>U</math>}} is a [[Continuous linear operator|continuous]] [[Linear form|linear functional]] on <math>C_c^\infty(U).</math> Said differently, a distribution on <math>U</math> is an element of the [[continuous dual space]] of <math>C_c^\infty(U)</math> when <math>C_c^\infty(U)</math> is endowed with its canonical LF topology.}}

There is a canonical [[Dual system|duality pairing]] between a distribution <math>T</math> on <math>U</math> and a test function <math>f \in C_c^\infty(U),</math> which is denoted using [[angle brackets]] by
<math display=block>\begin{cases}
\mathcal{D}'(U) \times C_c^\infty(U) \to \R \\
(T, f) \mapsto \langle T, f \rangle := T(f)
\end{cases}</math>

One interprets this notation as the distribution <math>T</math> acting on the test function <math>f</math> to give a scalar, or symmetrically as the test function <math>f</math> acting on the distribution <math>T.</math>

====Characterizations of distributions====

'''Proposition.''' If <math>T</math> is a [[Linear form|linear functional]] on <math>C_c^\infty(U)</math> then the following are equivalent:

# {{mvar|T}} is a distribution;
# {{mvar|T}} is [[Continuous function|continuous]];
# {{mvar|T}} is [[Continuous function|continuous]] at the origin;
# {{mvar|T}} is [[Uniform continuity|uniformly continuous]];
# {{mvar|T}} is a [[bounded operator]];
# {{mvar|T}} is [[sequentially continuous]];
#* explicitly, for every sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> that converges in <math>C_c^\infty(U)</math> to some <math>f \in C_c^\infty(U),</math> <math display=inline>\lim_{i \to \infty} T\left(f_i\right) = T(f);</math><ref group="note">Even though the topology of <math>C_c^\infty(U)</math> is not metrizable, a linear functional on <math>C_c^\infty(U)</math> is continuous if and only if it is sequentially continuous.</ref>
# {{mvar|T}} is [[sequentially continuous]] at the origin; in other words, {{mvar|T}} maps null sequences<ref group=note name="Def null sequence">A '''{{em|null sequence}}''' is a sequence that converges to the origin.</ref> to null sequences;
#* explicitly, for every sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> that converges in <math>C_c^\infty(U)</math> to the origin (such a sequence is called a {{em|null sequence}}), <math display=inline>\lim_{i \to \infty} T\left(f_i\right) = 0;</math>
#* a {{em|null sequence}} is by definition any sequence that converges to the origin;
# {{mvar|T}} maps null sequences to bounded subsets;
#* explicitly, for every sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> that converges in <math>C_c^\infty(U)</math> to the origin, the sequence <math>\left(T\left(f_i\right)\right)_{i=1}^\infty</math> is bounded;
# {{mvar|T}} maps [[Mackey convergence|Mackey convergent]] null sequences to bounded subsets;
#* explicitly, for every Mackey convergent null sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U),</math> the sequence <math>\left(T\left(f_i\right)\right)_{i=1}^\infty</math> is bounded;
#* a sequence <math>f_{\bull} = \left(f_i\right)_{i=1}^\infty</math> is said to be {{em|[[Mackey convergence|Mackey convergent]] to the origin}} if there exists a divergent sequence <math>r_{\bull} = \left(r_i\right)_{i=1}^\infty \to \infty</math> of positive real numbers such that the sequence <math>\left(r_i f_i\right)_{i=1}^\infty</math> is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
# The kernel of {{mvar|T}} is a closed subspace of <math>C_c^\infty(U);</math>
# The graph of {{mvar|T}} is closed;
<!------ START: Removed information ----
* Note in particular that the following seminorms on <math>C^\infty(U)</math> (which recall were defined earlier) all restrict to continuous seminorms on <math>C_c^\infty(U)</math>: <math>\ q_{i,K}, \ r_{i,K}, \ r_{t,K},</math> and <math>s_{p,K},</math> where {{mvar|K}} is any compact subset of {{mvar|U}}, <math>i \geq 0</math> is an integer, and <math>p</math> is a multi-index. So to show that {{mvar|T}} is continuous, it {{em|suffices}} to show that the restriction to <math>C_c^\infty(U)</math> of one of these seminorms, call it <math>g,</math> satisfies <math>|T| \leq C g</math> for some <math>C > 0.</math>
---- END: Removed information ---->
# There exists a continuous seminorm <math>g</math> on <math>C_c^\infty(U)</math> such that <math>|T| \leq g;</math>
# There exists a constant <math>C > 0</math> and a finite subset <math>\{g_1, \ldots, g_m\} \subseteq \mathcal{P}</math> (where <math>\mathcal{P}</math> is any collection of continuous seminorms that defines the canonical LF topology on <math>C_c^\infty(U)</math>) such that <math>|T| \leq C(g_1 + \cdots + g_m);</math><ref group="note">If <math>\mathcal{P}</math> is also [[Directed set|directed]] under the usual function comparison then we can take the finite collection to consist of a single element.</ref>
# For every compact subset <math>K\subseteq U</math> there exist constants <math>C>0</math> and <math>N\in \N</math> such that for all <math>f \in C^\infty(K),</math>{{sfn|Trèves|2006|pp=222-223}} <math display=block>|T(f)| \leq C \sup \{|\partial^\alpha f(x)| : x \in U, |\alpha|\leq N\};</math>
# For every compact subset <math>K\subseteq U</math> there exist constants <math>C_K>0</math> and <math>N_K\in \N</math> such that for all <math>f \in C_c^\infty(U)</math> with [[#support of a function|support]] contained in <math>K,</math><ref name="Grubb 2009 page=14">See for example {{harvnb|Grubb|2009|page=14}}.</ref> <math display=block>|T(f)| \leq C_K \sup \{|\partial^\alpha f(x)| : x \in K, |\alpha|\leq N_K\};</math>
# For any compact subset <math>K\subseteq U</math> and any sequence <math>\{f_i\}_{i=1}^\infty</math> in <math>C^\infty(K),</math> if <math>\{\partial^p f_i\}_{i=1}^\infty</math> converges uniformly to zero for all [[multi-index|multi-indices]] <math>p,</math> then <math>T(f_i) \to 0;</math>

====Topology on the space of distributions and its relation to the weak-* topology====

The set of all distributions on <math>U</math> is the [[continuous dual space]] of <math>C_c^\infty(U),</math> which when endowed with the [[Strong topology (polar topology)|strong dual topology]] is denoted by <math>\mathcal{D}'(U).</math> Importantly, unless indicated otherwise, the topology on <math>\mathcal{D}'(U)</math> is the [[strong dual topology]]; if the topology is instead the [[weak-* topology]] then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes <math>\mathcal{D}'(U)</math> into a [[Complete topological vector space|complete]] [[nuclear space]], to name just a few of its desirable properties.

Neither <math>C_c^\infty(U)</math> nor its strong dual <math>\mathcal{D}'(U)</math> is a [[sequential space]] and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is {{em|not}} enough to fully/correctly define their topologies).
However, a {{em|sequence}} in <math>\mathcal{D}'(U)</math> converges in the strong dual topology if and only if it converges in the [[weak-* topology]] (this leads many authors to use pointwise convergence to {{em|define}} the convergence of a sequence of distributions; this is fine for sequences but this is {{em|not}} guaranteed to extend to the convergence of [[Net (mathematics)|nets]] of distributions because a net may converge pointwise but fail to converge in the strong dual topology).
More information about the topology that <math>\mathcal{D}'(U)</math> is endowed with can be found in the article on [[spaces of test functions and distributions]] and the articles on [[Polar topology|polar topologies]] and [[dual system]]s.

A [[Linear map|{{em|linear}} map]] from <math>\mathcal{D}'(U)</math> into another [[locally convex topological vector space]] (such as any [[normed space]]) is [[Continuous function (topology)|continuous]] if and only if it is [[sequentially continuous]] at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general [[topological space]]s (for example, that are not also locally convex [[topological vector space]]s). The same is true of maps from <math>C_c^\infty(U)</math> (more generally, this is true of maps from any locally convex [[bornological space]]).

==Localization of distributions==

There is no way to define the value of a distribution in <math>\mathcal{D}'(U)</math> at a particular point of {{mvar|U}}. However, as is the case with functions, distributions on {{mvar|U}} restrict to give distributions on open subsets of {{mvar|U}}. Furthermore, distributions are {{em|locally determined}} in the sense that a distribution on all of {{mvar|U}} can be assembled from a distribution on an open cover of {{mvar|U}} satisfying some compatibility conditions on the overlaps. Such a structure is known as a [[Sheaf (mathematics)|sheaf]].

===Extensions and restrictions to an open subset===

Let <math>V \subseteq U</math> be open subsets of <math>\R^n.</math> 
Every function <math>f \in \mathcal{D}(V)</math> can be {{em|extended by zero}} from its domain {{mvar|V}} to a function on {{mvar|U}} by setting it equal to <math>0</math> on the [[Complement (set theory)|complement]] <math>U \setminus V.</math> This extension is a smooth compactly supported function called the {{em|trivial extension of <math>f</math> to <math>U</math>}} and it will be denoted by <math>E_{VU} (f).</math>
This assignment <math>f \mapsto E_{VU} (f)</math> defines the {{em|trivial extension}} operator 
<math>E_{VU} : \mathcal{D}(V) \to \mathcal{D}(U),</math> 
which is a continuous injective linear map. It is used to canonically identify <math>\mathcal{D}(V)</math> as a [[vector subspace]] of <math>\mathcal{D}(U)</math> (although {{em|not}} as a [[topological subspace]]). 
Its transpose ([[#Transpose of a linear operator|explained here]]) 
<math display=block>\rho_{VU} := {}^{t}E_{VU} : \mathcal{D}'(U) \to \mathcal{D}'(V),</math> 
is called the '''{{em|{{visible anchor|restriction map|text=restriction to <math>V</math> of distributions in <math>U</math>}}}}'''{{sfn|Trèves|2006|pp=245-247}} and as the name suggests, the image <math>\rho_{VU}(T)</math> of a distribution <math>T \in \mathcal{D}'(U)</math> under this map is a distribution on <math>V</math> called the '''restriction of <math>T</math> to <math>V.</math>''' The [[#Transpose of a linear operator|defining condition]] of the restriction <math>\rho_{VU}(T)</math> is:
<math display=block>\langle \rho_{VU} T, \phi \rangle = \langle T, E_{VU} \phi \rangle \quad \text{ for all } \phi \in \mathcal{D}(V).</math>
If <math>V \neq U</math> then the (continuous injective linear) trivial extension map <math>E_{VU} : \mathcal{D}(V) \to \mathcal{D}(U)</math> is {{em|not}} a topological embedding (in other words, if this linear injection was used to identify <math>\mathcal{D}(V)</math> as a subset of <math>\mathcal{D}(U)</math> then <math>\mathcal{D}(V)</math>'s topology would [[Comparison of topologies|strictly finer]] than the [[subspace topology]] that <math>\mathcal{D}(U)</math> induces on it; importantly, it would {{em|not}} be a [[topological subspace]] since that requires equality of topologies) and its range is also {{em|not}} dense in its [[codomain]] <math>\mathcal{D}(U).</math>{{sfn|Trèves|2006|pp=245-247}} Consequently if <math>V \neq U</math> then [[#restriction map|the restriction mapping]] is neither injective nor surjective.{{sfn|Trèves|2006|pp=245-247}} A distribution <math>S \in \mathcal{D}'(V)</math> is said to be '''{{em|extendible to {{mvar|U}}}}''' if it belongs to the range of the transpose of <math>E_{VU}</math> and it is called '''{{em|extendible}}''' if it is extendable to <math>\R^n.</math>{{sfn|Trèves|2006|pp=245-247}}

Unless <math>U = V,</math> the restriction to {{mvar|V}} is neither [[injective]] nor [[surjective]]. Lack of surjectivity follows since distributions can blow up towards the boundary of {{mvar|V}}. For instance, if <math>U = \R</math> and <math>V = (0, 2),</math> then the distribution
<math display=block>T(x) = \sum_{n=1}^\infty n \, \delta\left(x-\frac{1}{n}\right)</math>
is in <math>\mathcal{D}'(V)</math> but admits no extension to <math>\mathcal{D}'(U).</math>

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

{{Math theorem
| name = Theorem{{sfn |Trèves|2006|pp=253-255}}
| math_statement = Let <math>(U_i)_{i \in I}</math> be a collection of open subsets of <math>\R^n.</math> For each <math>i \in I,</math> let <math>T_i \in \mathcal{D}'(U_i)</math> and suppose that for all <math>i, j \in I,</math> the restriction of <math>T_i</math> to <math>U_i \cap U_j</math> is equal to the restriction of <math>T_j</math> to <math>U_i \cap U_j</math> (note that both restrictions are elements of <math>\mathcal{D}'(U_i \cap U_j)</math>). Then there exists a unique <math display=inline>T \in \mathcal{D}'(\bigcup_{i \in I} U_i)</math> such that for all <math>i \in I,</math> the restriction of {{mvar|T}} to <math>U_i</math> is equal to <math>T_i.</math>
}}

Let {{mvar|V}} be an open subset of {{mvar|U}}. <math>T \in \mathcal{D}'(U)</math> is said to '''{{em|vanish in {{mvar|V}}}}''' if for all <math>f \in \mathcal{D}(U)</math> such that <math>\operatorname{supp}(f) \subseteq V</math> we have <math>Tf = 0.</math> {{mvar|T}} vanishes in {{mvar|V}} if and only if the restriction of {{mvar|T}} to {{mvar|V}} is equal to 0, or equivalently, if and only if {{mvar|T}} lies in the [[kernel (algebra)|kernel]] of the restriction map <math>\rho_{VU}.</math>

{{Math theorem
| name = Corollary{{sfn |Trèves|2006| pp=253-255}}
| math_statement = Let <math>(U_i)_{i \in I}</math> be a collection of open subsets of <math>\R^n</math> and let <math display=inline>T \in \mathcal{D}'(\bigcup_{i \in I} U_i).</math> <math>T = 0</math> if and only if for each <math>i \in I,</math> the restriction of {{mvar|T}} to <math>U_i</math> is equal to 0.
}}

{{Math theorem| name=Corollary{{sfn |Trèves|2006|pp=253-255}}| math_statement= The union of all open subsets of {{mvar|U}} in which a distribution {{mvar|T}} vanishes is an open subset of {{mvar|U}} in which {{mvar|T}} vanishes.}}

===Support of a distribution===

This last corollary implies that for every distribution {{mvar|T}} on {{mvar|U}}, there exists a unique largest subset {{mvar|V}} of {{mvar|U}} such that {{mvar|T}} vanishes in {{mvar|V}} (and does not vanish in any open subset of {{mvar|U}} that is not contained in {{mvar|V}}); the complement in {{mvar|U}} of this unique largest open subset is called {{em|the '''support''' of {{mvar|T}}}}.{{sfn|Trèves|2006|pp=253-255}} Thus
<math display=block> \operatorname{supp}(T) = U \setminus \bigcup \{V \mid \rho_{VU}T = 0\}.</math>

If <math>f</math> is a locally integrable function on {{mvar|U}} and if <math>D_f</math> is its associated distribution, then the support of <math>D_f</math> is the smallest closed subset of {{mvar|U}} in the complement of which <math>f</math> is [[almost everywhere]] equal to 0.{{sfn|Trèves|2006|pp=253-255}} If <math>f</math> is continuous, then the support of <math>D_f</math> is equal to the closure of the set of points in {{mvar|U}} at which <math>f</math> does not vanish.{{sfn|Trèves|2006| pp=253-255}} The support of the distribution associated with the [[Dirac measure]] at a point <math>x_0</math> is the set <math>\{x_0\}.</math>{{sfn|Trèves|2006|pp=253-255}} If the support of a test function <math>f</math> does not intersect the support of a distribution {{mvar|T}} then <math>Tf = 0.</math> A distribution {{mvar|T}} is 0 if and only if its support is empty. If <math>f \in C^\infty(U)</math> is identically 1 on some open set containing the support of a distribution {{mvar|T}} then <math>f T = T.</math> If the support of a distribution {{mvar|T}} is compact then it has finite order and there is a constant <math>C</math> and a non-negative integer <math>N</math> such that:{{sfn|Rudin|1991|pp=149-181}}
<math display=block>|T \phi| \leq C\|\phi\|_N := C \sup \left\{\left|\partial^\alpha \phi(x)\right| : x \in U, |\alpha| \leq N \right\} \quad \text{ for all } \phi \in \mathcal{D}(U).</math>

If {{mvar|T}} has compact support, then it has a unique extension to a continuous linear functional <math>\widehat{T}</math> on <math>C^\infty(U)</math>; this function can be defined by <math>\widehat{T} (f) := T(\psi f),</math> where <math>\psi \in \mathcal{D}(U)</math> is any function that is identically 1 on an open set containing the support of {{mvar|T}}.{{sfn|Rudin|1991|pp=149-181}}

If <math>S, T \in \mathcal{D}'(U)</math> and <math>\lambda \neq 0</math> then <math>\operatorname{supp}(S + T) \subseteq \operatorname{supp}(S) \cup \operatorname{supp}(T)</math> and <math>\operatorname{supp}(\lambda T) = \operatorname{supp}(T).</math> Thus, distributions with support in a given subset <math>A \subseteq U</math> form a vector subspace of <math>\mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=255-257}} Furthermore, if <math>P</math> is a differential operator in {{mvar|U}}, then for all distributions {{mvar|T}} on {{mvar|U}} and all <math>f \in C^\infty(U)</math> we have <math>\operatorname{supp} (P(x, \partial) T) \subseteq \operatorname{supp}(T)</math> and <math>\operatorname{supp}(fT) \subseteq \operatorname{supp}(f) \cap \operatorname{supp}(T).</math>{{sfn|Trèves|2006|pp=255-257}}

===Distributions with compact support===

====Support in a point set and Dirac measures====

For any <math>x \in U,</math> let <math>\delta_x \in \mathcal{D}'(U)</math> denote the distribution induced by the Dirac measure at <math>x.</math> For any <math>x_0 \in U</math> and distribution <math>T \in \mathcal{D}'(U),</math> the support of {{mvar|T}} is contained in <math>\{x_0\}</math> if and only if {{mvar|T}} is a finite linear combination of derivatives of the Dirac measure at <math>x_0.</math>{{sfn|Trèves|2006|pp=264-266}} If in addition the order of {{mvar|T}} is <math>\leq k</math> then there exist constants <math>\alpha_p</math> such that:{{sfn|Rudin|1991|p=165}}
<math display=block>T = \sum_{|p| \leq k} \alpha_p \partial^p \delta_{x_0}.</math>

Said differently, if {{mvar|T}} has support at a single point <math>\{P\},</math> then {{mvar|T}} is in fact a finite linear combination of distributional derivatives of the <math>\delta</math> function at {{mvar|P}}. That is, there exists an integer {{mvar|m}} and complex constants <math>a_\alpha</math> such that
<math display=block>T = \sum_{|\alpha|\leq m} a_\alpha \partial^\alpha(\tau_P\delta)</math>
where <math>\tau_P</math> is the translation operator.

====Distribution with compact support====

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=149-181}}|math_statement=
Suppose {{mvar|T}} is a distribution on {{mvar|U}} with compact support {{mvar|K}}. There exists a continuous function <math>f</math> defined on {{mvar|U}} and a multi-index {{math|1=''p''}} such that
<math display=block>T = \partial^p f,</math>
where the derivatives are understood in the sense of distributions. That is, for all test functions <math>\phi</math> on {{mvar|U}},
<math display=block>T \phi = (-1)^{|p|} \int_{U} f(x) (\partial^p \phi)(x) \, dx.</math>
}}

====Distributions of finite order with support in an open subset====

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=149-181}}|math_statement=
Suppose {{mvar|T}} is a distribution on {{mvar|U}} with compact support {{mvar|K}} and let {{mvar|V}} be an open subset of {{mvar|U}} containing {{mvar|K}}. Since every distribution with compact support has finite order, take {{mvar|N}} to be the order of {{mvar|T}} and define <math>P:=\{0,1,\ldots, N+2\}^n.</math> There exists a family of continuous functions <math>(f_p)_{p\in P}</math> defined on {{mvar|U}} '''with support in {{mvar|V}}''' such that
<math display=block>T = \sum_{p \in P} \partial^p f_p,</math>
where the derivatives are understood in the sense of distributions. That is, for all test functions <math>\phi</math> on {{mvar|U}},
<math display=block>T \phi = \sum_{p \in P} (-1)^{|p|} \int_U f_p(x) (\partial^p \phi)(x) \, dx.</math>
}}

===Global structure of distributions===

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of <math>\mathcal{D}(U)</math> (or the [[Schwartz space]] <math>\mathcal{S}(\R^n)</math> 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 [[Sheaf (mathematics)|sheaves]]====

{{Math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=
Let {{mvar|T}} be a distribution on {{mvar|U}}.
There exists a sequence <math>(T_i)_{i=1}^\infty</math> in <math>\mathcal{D}'(U)</math> such that each {{mvar|T<sub>i</sub>}} has compact support and every compact subset <math>K \subseteq U</math> intersects the support of only finitely many <math>T_i,</math> and the sequence of partial sums <math>(S_j)_{j=1}^\infty,</math> defined by <math>S_j := T_1 + \cdots + T_j,</math> converges in <math>\mathcal{D}'(U)</math> to {{mvar|T}}; in other words we have:
<math display=block>T = \sum_{i=1}^\infty T_i.</math>
Recall that a sequence converges in <math>\mathcal{D}'(U)</math> (with its strong dual topology) if and only if it converges pointwise.
}}

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

By combining the above results, one may express any distribution on {{mvar|U}} as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on {{mvar|U}}. In other words, for arbitrary <math>T \in \mathcal{D}'(U)</math> we can write:
<math display=block>T = \sum_{i=1}^\infty \sum_{p \in P_i} \partial^p f_{ip},</math>
where <math>P_1, P_2, \ldots</math> are finite sets of multi-indices and the functions <math>f_{ip}</math> are continuous.

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=169-170}}|math_statement=
Let {{mvar|T}} be a distribution on {{mvar|U}}. For every multi-index {{mvar|p}} there exists a continuous function <math>g_p</math> on {{mvar|U}} such that

# any compact subset {{mvar|K}} of {{mvar|U}} intersects the support of only finitely many <math>g_p,</math> and
# <math>T = \sum\nolimits_p \partial^p g_p.</math>

Moreover, if {{mvar|T}} has finite order, then one can choose <math>g_p</math> in such a way that only finitely many of them are non-zero.
}}

Note that the infinite sum above is well-defined as a distribution. The value of {{mvar|T}} for a given <math>f \in \mathcal{D}(U)</math> can be computed using the finitely many <math>g_\alpha</math> that intersect the support of <math>f.</math>

==Operations on distributions==

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if <math>A:\mathcal{D}(U)\to\mathcal{D}(U)</math> is a linear map that is continuous with respect to the [[weak topology]], then it is not always possible to extend <math>A</math> to a map <math>A': \mathcal{D}'(U)\to \mathcal{D}'(U)</math> by classic extension theorems of topology or linear functional analysis.<ref group="note">The extension theorem for mappings  defined from a subspace S of a topological vector space E to the topological space E itself works for non-linear mappings as well, provided they are assumed to be [[uniformly continuous]]. But, unfortunately, this is not our case, we would desire to “extend” a linear continuous mapping A from a tvs E into another tvs F, in order to obtain a linear continuous mapping from the dual E’ to the dual F’ (note the order of spaces). In general, this is not even an extension problem, because (in general) E is not necessarily a subset of its own dual E’. Moreover, It is not a classic topological transpose problem, because the transpose of A goes from F’ to E’ and not from E’ to F’. Our case needs, indeed, a new order of ideas, involving the specific topological properties of the Laurent Schwartz spaces D(U) and D’(U), together with the fundamental concept of weak (or Schwartz) adjoint of the linear continuous operator A.</ref> The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that  
<math>
\langle Af,g\rangle = \langle f,Bg\rangle
</math>,
for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B. {{citation needed|date=September 2020}}<ref>{{Cite book |last=Strichartz |first=Robert |title=A Guide to Distribution Theory and Fourier Transforms |year=1993 |location=USA |pages=17 |language=English}}</ref>{{clarify|date=September 2020}}

===Preliminaries: Transpose of a linear operator===
{{anchor|Transpose of a linear operator}}
{{Main|Transpose of a linear map}}

Operations on distributions and spaces of distributions are often defined using the [[Transpose of a linear map|transpose]] of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in [[functional analysis]].<ref>{{harvnb|Strichartz|1994|loc=§2.3}}; {{harvnb|Trèves|2006}}.</ref> For instance, the well-known [[Hermitian adjoint]] of a linear operator between [[Hilbert space]]s is just the operator's transpose (but with the [[Riesz representation theorem]] used to identify each Hilbert space with its [[Strong dual space|continuous dual space]]). In general, the transpose of a continuous linear map <math>A : X \to Y</math> is the linear map 
<math display=block>{}^{t}A : Y' \to X' \qquad \text{ defined by } \qquad {}^{t}A(y') := y' \circ A,</math> 
or equivalently, it is the unique map satisfying <math>\langle y', A(x)\rangle = \left\langle {}^{t}A (y'), x \right\rangle</math> for all <math>x \in X</math> and all <math>y' \in Y'</math> (the prime symbol in <math>y'</math> does not denote a derivative of any kind; it merely indicates that <math>y'</math> is an element of the continuous dual space <math>Y'</math>). Since <math>A</math> is continuous, the transpose <math>{}^{t}A : Y' \to X'</math> is also continuous when both duals are endowed with their respective [[Strong dual space|strong dual topologies]]; it is also continuous when both duals are endowed with their respective [[Weak* topology|weak* topologies]] (see the articles [[Polar topology#Polar topologies and topological vector spaces|polar topology]] and [[Dual system#Weak topology|dual system]] for more details).

In the context of distributions, the characterization of the transpose can be refined slightly. Let <math>A : \mathcal{D}(U) \to \mathcal{D}(U)</math> be a continuous linear map. Then by definition, the transpose of <math>A</math> is the unique linear operator <math>{}^tA : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> that satisfies:
<math display=block>\langle {}^{t}A(T), \phi \rangle = \langle T, A(\phi) \rangle \quad \text{ for all } \phi \in \mathcal{D}(U) \text{ and all } T \in \mathcal{D}'(U).</math>

Since <math>\mathcal{D}(U)</math> is dense in <math>\mathcal{D}'(U)</math> (here, <math>\mathcal{D}(U)</math> actually refers to the set of distributions <math>\left\{D_\psi : \psi \in \mathcal{D}(U)\right\}</math>) it is sufficient that the defining equality hold for all distributions of the form <math>T = D_\psi</math> where <math>\psi \in \mathcal{D}(U).</math> Explicitly, this means that a continuous linear map <math>B : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> is equal to <math>{}^{t}A</math> if and only if the condition below holds:
<math display=block>\langle B(D_\psi), \phi \rangle = \langle {}^{t}A(D_\psi), \phi \rangle \quad \text{ for all } \phi, \psi \in \mathcal{D}(U)</math>
where the right-hand side equals <math>\langle {}^{t}A(D_\psi), \phi \rangle = \langle D_\psi, A(\phi) \rangle = \langle \psi, A(\phi) \rangle = \int_U \psi \cdot A(\phi) \,dx.</math>

===Differential operators===

====Differentiation of distributions====

Let <math>A : \mathcal{D}(U) \to \mathcal{D}(U)</math> be the partial derivative operator <math>\tfrac{\partial}{\partial x_k}.</math> To extend <math>A</math> we compute its transpose:
<math display=block>\begin{align}
\langle {}^{t}A(D_\psi), \phi \rangle
&= \int_U \psi (A\phi) \,dx && \text{(See above.)} \\
&= \int_U \psi \frac{\partial\phi}{\partial x_k} \, dx \\[4pt]
&= -\int_U \phi \frac{\partial\psi}{\partial x_k}\, dx && \text{(integration by parts)} \\[4pt]
&= -\left\langle \frac{\partial\psi}{\partial x_k}, \phi \right\rangle \\[4pt]
&= -\langle A \psi, \phi \rangle = \langle - A \psi, \phi \rangle
\end{align}</math>

Therefore <math>{}^{t}A = -A.</math> Thus, the partial derivative of <math>T</math> with respect to the coordinate <math>x_k</math> is defined by the formula
<math display=block>\left\langle \frac{\partial T}{\partial x_k}, \phi \right\rangle = - \left\langle T, \frac{\partial \phi}{\partial x_k} \right\rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

With this definition, every distribution is infinitely differentiable, and the derivative in the direction <math>x_k</math> is a [[linear operator]] on <math>\mathcal{D}'(U).</math>

More generally, if <math>\alpha</math> is an arbitrary [[multi-index]], then the partial derivative <math>\partial^\alpha T</math> of the distribution <math>T \in \mathcal{D}'(U)</math> is defined by
<math display=block>\langle \partial^\alpha T, \phi \rangle = (-1)^{|\alpha|} \langle T, \partial^\alpha \phi \rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

Differentiation of distributions is a continuous operator on <math>\mathcal{D}'(U);</math> this is an important and desirable property that is not shared by most other notions of differentiation.

If <math>T</math> is a distribution in <math>\R</math> then
<math display=block>\lim_{x \to 0} \frac{T - \tau_x T}{x} = T'\in \mathcal{D}'(\R),</math>
where <math>T'</math> is the derivative of <math>T</math> and <math>\tau_x</math> is a translation by <math>x;</math> thus the derivative of <math>T</math> may be viewed as a limit of quotients.{{sfn|Rudin|1991|p=180}}

====Differential operators acting on smooth functions====

A linear differential operator in <math>U</math> with smooth coefficients acts on the space of smooth functions on <math>U.</math> Given such an operator
<math display=inline>P := \sum_\alpha c_\alpha \partial^\alpha,</math>
we would like to define a continuous linear map, <math>D_P</math> that extends the action of <math>P</math> on <math>C^\infty(U)</math> to distributions on <math>U.</math> In other words, we would like to define <math>D_P</math> such that the following diagram [[Commutative diagram|commutes]]:
<math display=block>\begin{matrix}
\mathcal{D}'(U) & \stackrel{D_P}{\longrightarrow} & \mathcal{D}'(U) \\[2pt]
\uparrow & & \uparrow \\[2pt]
C^\infty(U) & \stackrel{P}{\longrightarrow} & C^\infty(U)
\end{matrix}</math>
where the vertical maps are given by assigning <math>f \in C^\infty(U)</math> its canonical distribution <math>D_f \in \mathcal{D}'(U),</math> which is defined by: 
<math display=block>D_f(\phi) = \langle f, \phi \rangle := \int_U f(x) \phi(x) \,dx \quad \text{ for all } \phi \in \mathcal{D}(U).</math> 
With this notation, the diagram commuting is equivalent to:
<math display=block>D_{P(f)} = D_PD_f \qquad \text{ for all } f \in C^\infty(U).</math>

To find <math>D_P,</math> the transpose <math>{}^{t} P : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> of the continuous induced map <math>P : \mathcal{D}(U)\to \mathcal{D}(U)</math> defined by <math>\phi \mapsto P(\phi)</math> is considered in the lemma below. 
This leads to the following definition of the differential operator on <math>U</math> called {{em|the '''formal transpose''' of <math>P,</math>}} which will be denoted by <math>P_*</math> to avoid confusion with the transpose map, that is defined by
<math display=block>P_* := \sum_\alpha b_\alpha \partial^\alpha \quad \text{ where } \quad b_\alpha := \sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \partial^{\beta-\alpha} c_\beta.</math>

{{math theorem|name=Lemma|math_statement= Let <math>P</math> be a linear differential operator with smooth coefficients in <math>U.</math> Then for all <math>\phi \in \mathcal{D}(U)</math> we have
<math display=block>\left\langle {}^{t}P(D_f), \phi \right\rangle = \left\langle D_{P_*(f)}, \phi \right\rangle,</math>
which is equivalent to:
<math display=block>{}^{t}P(D_f) = D_{P_*(f)}.</math>}}

{{collapse top|title=Proof|left=true}}
As discussed above, for any <math>\phi \in \mathcal{D}(U),</math> the transpose may be calculated by:
<math display=block>\begin{align}
\left\langle {}^{t}P(D_f), \phi \right\rangle &= \int_U f(x) P(\phi)(x) \,dx \\
&= \int_U f(x) \left[\sum\nolimits_\alpha c_\alpha(x) (\partial^\alpha \phi)(x) \right] \,dx \\
&= \sum\nolimits_\alpha \int_U f(x) c_\alpha(x) (\partial^\alpha \phi)(x) \,dx \\
&= \sum\nolimits_\alpha (-1)^{|\alpha|} \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,d x
\end{align}</math>

For the last line we used [[integration by parts]] combined with the fact that <math>\phi</math> and therefore all the functions <math>f (x)c_\alpha (x) \partial^\alpha \phi(x)</math> have compact support.<ref group="note">For example, let <math>U = \R</math> and take <math>P</math> to be the ordinary derivative for functions of one real variable and assume the support of <math>\phi</math> to be contained in the finite interval <math>(a,b),</math> then since <math>\operatorname{supp}(\phi) \subseteq (a, b)</math>
<math display=block>\begin{align}
\int_\R \phi'(x)f(x)\,dx &= \int_a^b \phi'(x)f(x) \,dx \\
&= \phi(x)f(x)\big\vert_a^b - \int_a^b f'(x) \phi(x) \,d x \\
&= \phi(b)f(b) - \phi(a)f(a) - \int_a^b f'(x) \phi(x) \,d x \\
&=-\int_a^b f'(x) \phi(x) \,d x
\end{align}</math>
where the last equality is because <math>\phi(a) = \phi(b) = 0.</math></ref> Continuing the calculation above, for all <math>\phi \in \mathcal{D}(U):</math>
<math display=block>\begin{align}
\left\langle {}^{t}P(D_f), \phi \right\rangle &=\sum\nolimits_\alpha (-1)^{|\alpha|} \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,dx && \text{As shown above} \\[4pt]
&= \int_U \phi(x) \sum\nolimits_\alpha (-1)^{|\alpha|} (\partial^\alpha(c_\alpha f))(x)\,dx \\[4pt]
&= \int_U \phi(x) \sum_\alpha \left[\sum_{\gamma \le \alpha} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x) \right] \,dx && \text{Leibniz rule}\\
&= \int_U \phi(x) \left[\sum_\alpha \sum_{\gamma \le \alpha} (-1)^{|\alpha|} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x)\right] \,dx \\
&= \int_U \phi(x) \left[ \sum_\alpha \left[ \sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \left(\partial^{\beta-\alpha}c_{\beta}\right)(x) \right] (\partial^\alpha f)(x)\right] \,dx && \text{Grouping terms by derivatives of } f \\
&= \int_U \phi(x) \left[\sum\nolimits_\alpha b_\alpha(x) (\partial^\alpha f)(x) \right] \, dx && b_\alpha:=\sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \partial^{\beta-\alpha}c_{\beta} \\
&= \left\langle \left(\sum\nolimits_\alpha b_\alpha \partial^\alpha \right) (f), \phi \right\rangle
\end{align}</math>
{{collapse bottom}}

The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, <math>P_{**}= P,</math>{{sfn|Trèves|2006|pp=247-252}} enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator <math>P_* : C_c^\infty(U) \to C_c^\infty(U)</math> defined by <math>\phi \mapsto P_*(\phi).</math> We claim that the transpose of this map, <math>{}^{t}P_* : \mathcal{D}'(U) \to \mathcal{D}'(U),</math> can be taken as <math>D_P.</math> To see this, for every <math>\phi \in \mathcal{D}(U),</math> compute its action on a distribution of the form <math>D_f</math> with <math>f \in C^\infty(U)</math>:

<math display=block>\begin{align}
\left\langle {}^{t}P_*\left(D_f\right),\phi \right\rangle &= \left\langle D_{P_{**}(f)}, \phi \right\rangle && \text{Using Lemma above with } P_* \text{ in place of } P\\
&= \left\langle D_{P(f)}, \phi \right\rangle && P_{**} = P
\end{align}</math>

We call the continuous linear operator <math>D_P := {}^{t}P_* : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> the '''{{em|differential operator on distributions extending <math>P</math>}}'''.{{sfn|Trèves|2006|pp=247-252}} Its action on an arbitrary distribution <math>S</math> is defined via:
<math display=block>D_P(S)(\phi) = S\left(P_*(\phi)\right) \quad \text{ for all } \phi \in \mathcal{D}(U).</math>

If <math>(T_i)_{i=1}^\infty</math> converges to <math>T \in \mathcal{D}'(U)</math> then for every multi-index <math>\alpha, (\partial^\alpha T_i)_{i=1}^\infty</math> converges to <math>\partial^\alpha T \in \mathcal{D}'(U).</math>

====Multiplication of distributions by smooth functions====

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if <math>f</math> is a smooth function then <math>P := f(x)</math> is a differential operator of order 0, whose formal transpose is itself (that is, <math>P_* = P</math>). The induced differential operator <math>D_P : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> maps a distribution <math>T</math> to a distribution denoted by <math>fT := D_P(T).</math> We have thus defined the multiplication of a distribution by a smooth function.

We now give an alternative presentation of the multiplication of a distribution <math>T</math> on <math>U</math> by a smooth function <math>m : U \to \R.</math> The product <math>mT</math> is defined by
<math display=block>\langle mT, \phi \rangle = \langle T, m\phi \rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

This definition coincides with the transpose definition since if <math>M : \mathcal{D}(U) \to \mathcal{D}(U)</math> is the operator of multiplication by the function <math>m</math> (that is, <math>(M\phi)(x) = m(x)\phi(x)</math>), then
<math display=block>\int_U (M \phi)(x) \psi(x)\,dx = \int_U m(x) \phi(x) \psi(x)\,d x = \int_U \phi(x) m(x) \psi(x) \,d x = \int_U \phi(x) (M \psi)(x)\,d x,</math>
so that <math>{}^tM = M.</math>

Under multiplication by smooth functions, <math>\mathcal{D}'(U)</math> is a [[Module (mathematics)|module]] over the [[ring (mathematics)|ring]] <math>C^\infty(U).</math> With this definition of multiplication by a smooth function, the ordinary [[product rule]] of calculus remains valid. However, some unusual identities also arise. For example, if <math>\delta</math> is the Dirac delta distribution on <math>\R,</math> then <math>m \delta = m(0) \delta,</math> and if <math>\delta^'</math> is the derivative of the delta distribution, then
<math display=block>m\delta' = m(0) \delta' - m' \delta = m(0) \delta' - m'(0) \delta.</math>

The bilinear multiplication map <math>C^\infty(\R^n) \times \mathcal{D}'(\R^n) \to \mathcal{D}'\left(\R^n\right)</math> given by <math>(f,T) \mapsto fT</math> is {{em|not}} continuous; it is however, [[hypocontinuous]].{{sfn|Trèves|2006|p=423}}

'''Example.''' The product of any distribution <math>T</math> with the function that is identically {{math|1}} on <math>U</math> is equal to <math>T.</math>

'''Example.''' Suppose <math>(f_i)_{i=1}^\infty</math> is a sequence of test functions on <math>U</math> that converges to the constant function <math>1 \in C^\infty(U).</math> For any distribution <math>T</math> on <math>U,</math> the sequence <math>(f_i T)_{i=1}^\infty</math> converges to <math>T \in \mathcal{D}'(U).</math>{{sfn|Trèves|2006|p=261}}

If <math>(T_i)_{i=1}^\infty</math> converges to <math>T \in \mathcal{D}'(U)</math> and <math>(f_i)_{i=1}^\infty</math> converges to <math>f \in C^\infty(U)</math> then <math>(f_i T_i)_{i=1}^\infty</math> converges to <math>fT \in \mathcal{D}'(U).</math>

=====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 support]]s are disjoint.<ref name="StackOverflow">{{cite web|url=https://math.stackexchange.com/q/2338283|title=Multiplication of two distributions whose singular supports are disjoint|date=Jun 27, 2017|publisher=Stack Exchange Network|author=Per Persson (username: md2perpe)}}</ref> With more effort, it is possible to define a well-behaved product of several distributions provided their [[wave front set]]s 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 <math>\operatorname{p.v.} \frac{1}{x}</math> is the distribution obtained by the [[Cauchy principal value]]
<math display=block>\left(\operatorname{p.v.} \frac{1}{x}\right)(\phi) = \lim_{\varepsilon\to 0^+} \int_{|x| \geq \varepsilon} \frac{\phi(x)}{x}\, dx \quad \text{ for all } \phi \in \mathcal{S}(\R).</math>

If <math>\delta</math> is the Dirac delta distribution then
<math display=block>(\delta \times x) \times \operatorname{p.v.} \frac{1}{x} = 0</math>
but,
<math display=block>\delta \times \left(x \times \operatorname{p.v.} \frac{1}{x}\right) = \delta</math>
so the product of a distribution by a smooth function (which is always well-defined) cannot be extended to an [[Associativity|associative]] product on the space of distributions.

Thus, nonlinear problems cannot be posed in general and thus are 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 (physics)|regularization]] of [[Ultraviolet divergence|divergences]]. Here [[Henri Epstein]] and [[Vladimir Glaser]] developed the mathematically rigorous (but extremely technical) {{em|[[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{{Citation needed|reason=Why are they not satisfactory?|date=July 2019}} theories of [[Algebra (ring theory)|algebra]]s of [[generalized function]]s have been developed, among which [[Colombeau algebra|Colombeau's (simplified) algebra]] is maybe the most popular in use today.

Inspired by Lyons' [[rough path]] theory,<ref>{{Cite journal|last1=Lyons|first1=T.|title=Differential equations driven by rough signals|doi=10.4171/RMI/240|journal=Revista Matemática Iberoamericana|pages=215–310|year=1998|volume=14 |issue=2 |doi-access=free}}</ref> [[Martin Hairer]] proposed a consistent way of multiplying distributions with certain structures ([[regularity structures]]<ref>{{cite journal|last1=Hairer|first1=Martin|title=A theory of regularity structures|journal=Inventiones Mathematicae|date=2014|doi=10.1007/s00222-014-0505-4|volume=198|issue=2|pages=269–504|bibcode=2014InMat.198..269H|arxiv=1303.5113|s2cid=119138901 }}</ref>), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on [[Jean-Michel Bony|Bony]]'s [[paraproduct]] from Fourier analysis.

===Composition with a smooth function===

Let <math>T</math> be a distribution on <math>U.</math> Let <math>V</math> be an open set in <math>\R^n</math> and <math>F : V \to U.</math> If <math>F</math> is a [[Submersion (mathematics)|submersion]] then it is possible to define
<math display=block>T \circ F \in \mathcal{D}'(V).</math>

This is {{em|the '''composition''' of the distribution <math>T</math> with <math>F</math>}}, and is also called {{em|the '''[[Pullback (differential geometry)|pullback]]''' of <math>T</math> along <math>F</math>}}, sometimes written
<math display=block>F^\sharp : T \mapsto F^\sharp T = T \circ F.</math>

The pullback is often denoted <math>F^*,</math> although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.

The condition that <math>F</math> be a submersion is equivalent to the requirement that the [[Jacobian matrix and determinant|Jacobian]] derivative <math>d F(x)</math> of <math>F</math> is a [[surjective]] linear map for every <math>x \in V.</math> A necessary (but not sufficient) condition for extending <math>F^{\#}</math> to distributions is that <math>F</math> be an [[open mapping]].<ref>See for example {{harvnb|Hörmander|1983|loc=Theorem 6.1.1}}.</ref> The [[Inverse function theorem]] ensures that a submersion satisfies this condition.

If <math>F</math> is a submersion, then <math>F^{\#}</math> is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since <math>F^{\#}</math> is a continuous linear operator on <math>\mathcal{D}(U).</math> Existence, however, requires using the [[Integration by substitution|change of variables]] formula, the inverse function theorem (locally), and a [[partition of unity]] argument.<ref>See {{harvnb|Hörmander|1983|loc=Theorem 6.1.2}}.</ref>

In the special case when <math>F</math> is a [[diffeomorphism]] from an open subset <math>V</math> of <math>\R^n</math> onto an open subset <math>U</math> of <math>\R^n</math> change of variables under the integral gives:
<math display=block>\int_V \phi\circ F(x) \psi(x)\,dx = \int_U \phi(x) \psi \left(F^{-1}(x) \right) \left|\det dF^{-1}(x) \right|\,dx.</math>

In this particular case, then, <math>F^{\#}</math> is defined by the transpose formula:
<math display=block>\left\langle F^\sharp T, \phi \right\rangle = \left\langle T, \left|\det d(F^{-1}) \right|\phi\circ F^{-1} \right\rangle.</math>

===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 <math>f</math> and <math>g</math> are functions on <math>\R^n</math> then we denote by <math>f\ast g</math> {{em|the '''convolution''' of <math>f</math> and <math>g,</math>}} defined at <math>x \in \R^n</math> to be the integral
<math display=block>(f \ast g)(x) := \int_{\R^n} f(x-y) g(y) \,dy = \int_{\R^n} f(y)g(x-y) \,dy</math>
provided that the integral exists. If <math>1 \leq p, q, r \leq \infty</math> are such that <math display=inline>\frac{1}{r} = \frac{1}{p} + \frac{1}{q} - 1</math> then for any functions <math>f \in L^p(\R^n)</math> and <math>g \in L^q(\R^n)</math> we have <math>f \ast g \in L^r(\R^n)</math> and <math>\|f\ast g\|_{L^r} \leq \|f\|_{L^p} \|g\|_{L^q}.</math>{{sfn|Trèves|2006|pp=278-283}} If <math>f</math> and <math>g</math> are continuous functions on <math>\R^n,</math> at least one of which has compact support, then <math>\operatorname{supp}(f \ast g) \subseteq \operatorname{supp} (f) + \operatorname{supp} (g)</math> and if <math>A\subseteq \R^n</math> then the value of <math>f\ast g</math> on <math>A</math> do {{em|not}} depend on the values of <math>f</math> outside of the [[Minkowski sum]] <math>A -\operatorname{supp} (g) = \{a-s : a\in A, s\in \operatorname{supp}(g)\}.</math>{{sfn|Trèves|2006|pp=278-283}}

Importantly, if <math>g \in L^1(\R^n)</math> has compact support then for any <math>0 \leq k \leq \infty,</math> the convolution map <math>f \mapsto f \ast g</math> is continuous when considered as the map <math>C^k(\R^n) \to C^k(\R^n)</math> or as the map <math>C_c^k(\R^n) \to C_c^k(\R^n).</math>{{sfn|Trèves|2006|pp=278-283}}

====Translation and symmetry====

Given <math>a \in \R^n,</math> the translation operator <math>\tau_a</math> sends <math>f : \R^n \to \Complex</math> to <math>\tau_a f : \R^n \to \Complex,</math> defined by <math>\tau_a f(y) = f(y-a).</math> This can be extended by the transpose to distributions in the following way: given a distribution <math>T,</math> {{em|the '''translation''' of <math>T</math> by <math>a</math>}} is the distribution <math>\tau_a T : \mathcal{D}(\R^n) \to \Complex</math> defined by <math>\tau_a T(\phi) := \left\langle T, \tau_{-a} \phi \right\rangle.</math>{{sfn|Trèves|2006|pp=284-297}}<ref>See for example {{harvnb|Rudin|1991|loc=§6.29}}.</ref>

Given <math>f : \R^n \to \Complex,</math> define the function <math>\tilde{f} : \R^n \to \Complex</math> by <math>\tilde{f}(x) := f(-x).</math> Given a distribution <math>T,</math> let <math>\tilde{T} : \mathcal{D}(\R^n) \to \Complex</math> be the distribution defined by <math>\tilde{T}(\phi) := T \left(\tilde{\phi}\right).</math> The operator <math>T \mapsto \tilde{T}</math> is called '''{{em|the symmetry with respect to the origin}}'''.{{sfn|Trèves|2006|pp=284-297}}

====Convolution of a test function with a distribution====

Convolution with <math>f \in \mathcal{D}(\R^n)</math> defines a linear map:
<math display=block>\begin{alignat}{4}
C_f : \,& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) \\
        & g                 && \mapsto\,&& f \ast g \\
\end{alignat}</math>
which is [[continuous function|continuous]] with respect to the canonical [[LF space]] topology on <math>\mathcal{D}(\R^n).</math>

Convolution of <math>f</math> with a distribution <math>T \in \mathcal{D}'(\R^n)</math> can be defined by taking the transpose of <math>C_f</math> relative to the duality pairing of <math>\mathcal{D}(\R^n)</math> with the space <math>\mathcal{D}'(\R^n)</math> of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} If <math>f, g, \phi \in \mathcal{D}(\R^n),</math> then by [[Fubini's theorem]]
<math display=block>\langle C_fg, \phi \rangle = \int_{\R^n}\phi(x)\int_{\R^n}f(x-y) g(y) \,dy \,dx = \left\langle g,C_{\tilde{f}}\phi \right\rangle.</math>

Extending by continuity, the convolution of <math>f</math> with a distribution <math>T</math> is defined by
<math display=block>\langle f \ast T, \phi \rangle = \left\langle T, \tilde{f} \ast \phi \right\rangle, \quad \text{ for all } \phi \in \mathcal{D}(\R^n).</math>

An alternative way to define the convolution of a test function <math>f</math> and a distribution <math>T</math> is to use the translation operator <math>\tau_a.</math> The convolution of the compactly supported function <math>f</math> and the distribution <math>T</math> is then the function defined for each <math>x \in \R^n</math> by
<math display=block>(f \ast T)(x) = \left\langle T, \tau_x \tilde{f} \right\rangle.</math>

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution <math>T</math> has compact support, and if <math>f</math> is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on <math>\Complex^n</math> to <math>\R^n,</math> the restriction of an entire function of exponential type in <math>\Complex^n</math> to <math>\R^n</math>), then the same is true of <math>T \ast f.</math>{{sfn|Trèves|2006|pp=284-297}} If the distribution <math>T</math> has compact support as well, then <math>f\ast T</math> is a compactly supported function, and the [[Titchmarsh convolution theorem]] {{harvtxt|Hörmander|1983|loc=Theorem 4.3.3}} implies that:
<math display=block>\operatorname{ch}(\operatorname{supp}(f \ast T)) = \operatorname{ch}(\operatorname{supp}(f)) + \operatorname{ch} (\operatorname{supp}(T))</math>
where <math>\operatorname{ch}</math> denotes the [[convex hull]] and <math>\operatorname{supp}</math> denotes the support.

====Convolution of a smooth function with a distribution====

Let <math>f \in C^\infty(\R^n)</math> and <math>T \in \mathcal{D}'(\R^n)</math> and assume that at least one of <math>f</math> and <math>T</math> has compact support. The '''{{em|convolution}}''' of <math>f</math> and <math>T,</math> denoted by <math>f \ast T</math> or by <math>T \ast f,</math> is the smooth function:{{sfn|Trèves|2006|pp=284-297}}
<math display=block>\begin{alignat}{4}
f \ast T : \,& \R^n && \to    \,&& \Complex \\
             & x    && \mapsto\,&& \left\langle T, \tau_x \tilde{f} \right\rangle \\
\end{alignat}</math>
satisfying for all <math>p \in \N^n</math>:
<math display=block>\begin{align}
&\operatorname{supp}(f \ast T) \subseteq \operatorname{supp}(f)+ \operatorname{supp}(T) \\[6pt]
&\text{ for all } p \in \N^n: \quad
\begin{cases}\partial^p \left\langle T, \tau_x \tilde{f} \right\rangle = \left\langle T, \partial^p \tau_x \tilde{f} \right\rangle \\
\partial^p (T \ast f) = (\partial^p T) \ast f = T \ast (\partial^p f).
\end{cases}
\end{align}</math>

Let <math>M</math> be the map <math>f \mapsto T \ast f</math>. If <math>T</math> is a distribution, then <math>M</math> is continuous as a map <math>\mathcal{D}(\R^n) \to C^\infty(\R^n)</math>. If <math>T</math> also has compact support, then <math>M</math> is also continuous as the map <math>C^\infty(\R^n) \to C^\infty(\R^n)</math> and continuous as the map <math>\mathcal{D}(\R^n) \to \mathcal{D}(\R^n).</math>{{sfn|Trèves|2006|pp=284-297}}

If <math>L : \mathcal{D}(\R^n) \to C^\infty(\R^n)</math> is a continuous linear map such that <math>L \partial^\alpha \phi = \partial^\alpha L \phi</math> for all <math>\alpha</math> and all <math>\phi \in \mathcal{D}(\R^n)</math> then there exists a distribution <math>T \in \mathcal{D}'(\R^n)</math> such that <math>L \phi = T \circ \phi</math> for all <math>\phi \in \mathcal{D}(\R^n).</math>{{sfn|Rudin|1991|pp=149-181}}

'''Example.'''{{sfn|Rudin|1991|pp=149-181}} Let <math>H</math> be the [[Heaviside step function|Heaviside function]] on <math>\R.</math> For any <math>\phi \in \mathcal{D}(\R),</math>
<math display=block>(H \ast \phi)(x) = \int_{-\infty}^x \phi(t) \, dt.</math>

Let <math>\delta</math> be the Dirac measure at 0 and let <math>\delta'</math> be its derivative as a distribution. Then <math>\delta' \ast H = \delta</math> and <math>1 \ast \delta' = 0.</math> Importantly, the associative law fails to hold:
<math display=block>1 = 1 \ast \delta = 1 \ast (\delta' \ast H ) \neq (1 \ast \delta') \ast H = 0 \ast H = 0.</math>

====Convolution of distributions====

It is also possible to define the convolution of two distributions <math>S</math> and <math>T</math> on <math>\R^n,</math> provided one of them has compact support. Informally, to define <math>S \ast T</math> where <math>T</math> has compact support, the idea is to extend the definition of the convolution <math>\,\ast\,</math> to a linear operation on distributions so that the associativity formula
<math display=block>S \ast (T \ast \phi) = (S \ast T) \ast \phi</math>
continues to hold for all test functions <math>\phi.</math><ref>{{harvnb|Hörmander|1983|loc=§IV.2}} proves the uniqueness of such an extension.</ref>

It is also possible to provide a more explicit characterization of the convolution of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} Suppose that <math>S</math> and <math>T</math> are distributions and that <math>S</math> has compact support. Then the linear maps
<math display=block>\begin{alignat}{9}
\bullet \ast \tilde{S} : \,& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) && \quad \text{ and } \quad && \bullet \ast \tilde{T} : \,&& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) \\
    & f                       && \mapsto\,&& f \ast \tilde{S}    &&       &&     && f                       && \mapsto\,&& f \ast \tilde{T} \\
\end{alignat}</math>
are continuous. The transposes of these maps:
<math display=block>{}^{t}\left(\bullet \ast \tilde{S}\right) : \mathcal{D}'(\R^n) \to \mathcal{D}'(\R^n) \qquad {}^{t}\left(\bullet \ast \tilde{T}\right) : \mathcal{E}'(\R^n) \to \mathcal{D}'(\R^n)</math>
are consequently continuous and it can also be shown that{{sfn|Trèves|2006|pp=284-297}}
<math display=block>{}^{t}\left(\bullet \ast \tilde{S}\right)(T) = {}^{t}\left(\bullet \ast \tilde{T}\right)(S).</math>

This common value is called {{em|the '''convolution''' of <math>S</math> and <math>T</math>}} and it is a distribution that is denoted by <math>S \ast T</math> or <math>T \ast S.</math> It satisfies <math>\operatorname{supp} (S \ast T) \subseteq \operatorname{supp}(S) + \operatorname{supp}(T).</math>{{sfn|Trèves|2006|pp=284-297}} If <math>S</math> and <math>T</math> are two distributions, at least one of which has compact support, then for any <math>a \in \R^n,</math> <math>\tau_a(S \ast T) = \left(\tau_a S\right) \ast T = S \ast \left(\tau_a T\right).</math>{{sfn|Trèves|2006|pp=284-297}} If <math>T</math> is a distribution in <math>\R^n</math> and if <math>\delta</math> is a [[Dirac measure]] then <math>T \ast \delta = T = \delta \ast T</math>;{{sfn|Trèves|2006|pp=284-297}} thus <math>\delta</math> is the [[identity element]] of the convolution operation. Moreover, if <math>f</math> is a function then <math>f \ast \delta^{\prime} = f^{\prime} = \delta^{\prime} \ast f</math> where now the associativity of convolution implies that <math>f^{\prime} \ast g = g^{\prime} \ast f</math> for all functions <math>f</math> and <math>g.</math>

Suppose that it is <math>T</math> that has compact support. For <math>\phi \in \mathcal{D}(\R^n)</math> consider the function
<math display=block>\psi(x) = \langle T, \tau_{-x} \phi \rangle.</math>

It can be readily shown that this defines a smooth function of <math>x,</math> which moreover has compact support. The convolution of <math>S</math> and <math>T</math> is defined by
<math display=block>\langle S \ast T, \phi \rangle = \langle S, \psi \rangle.</math>

This generalizes the classical notion of [[convolution]] of functions and is compatible with differentiation in the following sense: for every multi-index <math>\alpha.</math>
<math display=block>\partial^\alpha(S \ast T) = (\partial^\alpha S) \ast T = S \ast (\partial^\alpha T).</math>

The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is [[associative]].{{sfn|Trèves|2006|pp=284-297}}

This definition of convolution remains valid under less restrictive assumptions about <math>S</math> and <math>T.</math><ref>See for instance {{harvnb|Gel'fand|Shilov|1966–1968|loc=v. 1, pp. 103–104}} and {{harvnb|Benedetto|1997|loc=Definition 2.5.8}}.</ref>

The convolution of distributions with compact support induces a continuous bilinear map <math>\mathcal{E}' \times \mathcal{E}' \to \mathcal{E}'</math> defined by <math>(S,T) \mapsto S * T,</math> where <math>\mathcal{E}'</math> denotes the space of distributions with compact support.{{sfn|Trèves|2006|p=423}} However, the convolution map as a function <math>\mathcal{E}' \times \mathcal{D}' \to \mathcal{D}'</math> is {{em|not}} continuous{{sfn|Trèves|2006|p=423}} although it is separately continuous.{{sfn|Trèves|2006|p=294}} The convolution maps <math>\mathcal{D}(\R^n) \times \mathcal{D}' \to \mathcal{D}'</math> and <math>\mathcal{D}(\R^n) \times \mathcal{D}' \to \mathcal{D}(\R^n)</math> given by <math>(f, T) \mapsto f * T</math> both {{em|fail}} to be continuous.{{sfn|Trèves|2006|p=423}} Each of these non-continuous maps is, however, [[separately continuous]] and [[hypocontinuous]].{{sfn|Trèves|2006|p=423}}

====Convolution versus multiplication====

In general, [[Regularization (physics)|regularity]] is required for multiplication products, and [[Principle of locality|locality]] is required for convolution products. It is expressed in the following extension of the [[Convolution theorem|Convolution Theorem]] which guarantees the existence of both convolution and multiplication products. Let <math>F(\alpha) = f \in \mathcal{O}'_C</math> be a rapidly decreasing tempered distribution or, equivalently, <math>F(f) = \alpha \in \mathcal{O}_M</math> be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let <math>F</math> be the normalized (unitary, ordinary frequency) [[Fourier transform]].<ref>{{cite book|last=Folland|first=G.B.|title=Harmonic Analysis in Phase Space|publisher=Princeton University Press|publication-place=Princeton, NJ|year=1989}}</ref> Then, according to {{harvtxt|Schwartz|1951}},
<math display=block>F(f * g) = F(f) \cdot F(g) \qquad \text{ and } \qquad F(\alpha \cdot g) = F(\alpha) * F(g)</math>
hold within the space of tempered distributions.<ref>{{cite book|last=Horváth|first=John|author-link = John Horvath (mathematician)|title=Topological Vector Spaces and Distributions|publisher=Addison-Wesley Publishing Company|publication-place=Reading, MA|year=1966}}</ref><ref>{{cite book|last=Barros-Neto|first=José|title=An Introduction to the Theory of Distributions|publisher=Dekker|publication-place=New York, NY|year=1973}}</ref><ref>{{cite book|last=Petersen|first=Bent E.|title=Introduction to the Fourier Transform and Pseudo-Differential Operators|publisher=Pitman Publishing|publication-place=Boston, MA|year=1983}}</ref> In particular, these equations become the [[Poisson summation formula|Poisson Summation Formula]] if <math>g \equiv \operatorname{\text{Ш}}</math> is the [[Dirac comb|Dirac Comb]].<ref>{{cite book|last=Woodward|first=P.M.|title=Probability and Information Theory with Applications to Radar|publisher=Pergamon Press|publication-place=Oxford, UK|year=1953}}</ref> The space of all rapidly decreasing tempered distributions is also called the space of {{em|convolution operators}} <math>\mathcal{O}'_C</math> and the space of all ordinary functions within the space of tempered distributions is also called the space of {{em|multiplication operators}} <math>\mathcal{O}_M.</math> More generally, <math>F(\mathcal{O}'_C) = \mathcal{O}_M</math> and <math>F(\mathcal{O}_M) = \mathcal{O}'_C.</math>{{sfn|Trèves|2006|pp=318-319}}<ref>{{cite book|last1=Friedlander|first1=F.G.|last2=Joshi|first2=M.S.|title=Introduction to the Theory of Distributions|publisher=Cambridge University Press|publication-place=Cambridge, UK|year=1998}}</ref> A particular case is the [[Paley–Wiener theorem#Schwartz's Paley–Wiener theorem|Paley-Wiener-Schwartz Theorem]] which states that <math>F(\mathcal{E}') = \operatorname{PW}</math> and <math>F(\operatorname{PW} ) = \mathcal{E}'.</math> This is because <math>\mathcal{E}' \subseteq \mathcal{O}'_C</math> and <math>\operatorname{PW} \subseteq \mathcal{O}_M.</math> In other words, compactly supported tempered distributions <math>\mathcal{E}'</math> belong to the space of {{em|convolution operators}} <math>\mathcal{O}'_C</math> and
Paley-Wiener functions <math>\operatorname{PW},</math> better known as [[Bandlimiting|bandlimited functions]], belong to the space of {{em|multiplication operators}} <math>\mathcal{O}_M.</math>{{sfn|Schwartz|1951}}

For example, let <math>g \equiv \operatorname{\text{Ш}} \in \mathcal{S}'</math> be the Dirac comb and <math>f \equiv \delta \in \mathcal{E}'</math> be the [[Dirac delta function|Dirac delta]];then <math>\alpha \equiv 1 \in \operatorname{PW}</math> is the function that is constantly one and both equations yield the [[Dirac comb#Dirac-comb identity|Dirac-comb identity]]. Another example is to let <math>g</math> be the Dirac comb and <math>f \equiv \operatorname{rect} \in \mathcal{E}'</math> be the [[rectangular function]]; then <math>\alpha \equiv \operatorname{sinc} \in \operatorname{PW}</math> is the [[sinc function]] and both equations yield the [[Nyquist–Shannon sampling theorem|Classical Sampling Theorem]] for suitable <math>\operatorname{rect}</math> functions. More generally, if <math>g</math> is the Dirac comb and <math>f \in \mathcal{S} \subseteq \mathcal{O}'_C \cap \mathcal{O}_M</math> is a [[Smoothness|smooth]] [[window function]] ([[Schwartz space|Schwartz function]]), for example, the [[Gaussian function|Gaussian]], then <math>\alpha \in \mathcal{S}</math> is another smooth window function (Schwartz function). They are known as [[mollifier]]s, especially in [[partial differential equation]]s theory, or as [[Regularization (mathematics)|regularizers]] in [[Regularization (physics)|physics]] because they allow turning [[generalized function]]s into [[Function (mathematics)|regular functions]].

===Tensor products of distributions{{anchor|Tensor product of distributions}}===

Let <math>U \subseteq \R^m</math> and <math>V \subseteq \R^n</math> be open sets. Assume all vector spaces to be over the field <math>\mathbb{F},</math> where <math>\mathbb{F}=\R</math> or <math>\Complex.</math> For <math>f \in \mathcal{D}(U \times V)</math> define for every <math>u \in U</math> and every <math>v \in V</math> the following functions:
<math display=block>\begin{alignat}{9}
f_u : \,& V && \to    \,&& \mathbb{F} && \quad \text{ and } \quad && f^v : \,&& U && \to    \,&& \mathbb{F} \\
        & y && \mapsto\,&& f(u, y)    &&                          &&         && x && \mapsto\,&& f(x, v) \\
\end{alignat}</math>

Given <math>S \in \mathcal{D}^{\prime}(U)</math> and <math>T \in \mathcal{D}^{\prime}(V),</math> define the following functions:
<math display=block>\begin{alignat}{9}
\langle S, f^{\bullet}\rangle : \,& V && \to    \,&& \mathbb{F} && \quad \text{ and } \quad && \langle T, f_{\bullet}\rangle : \,&& U && \to    \,&& \mathbb{F} \\
                                  & v && \mapsto\,&& \langle S, f^v \rangle &&              &&                                   && u && \mapsto\,&& \langle T, f_u \rangle \\
\end{alignat}</math>
where <math>\langle T, f_{\bullet}\rangle \in \mathcal{D}(U)</math> and <math>\langle S, f^{\bullet}\rangle \in \mathcal{D}(V).</math> 
These definitions associate every <math>S \in \mathcal{D}'(U)</math> and <math>T \in \mathcal{D}'(V)</math> with the (respective) continuous linear map:
<math display=block>\begin{alignat}{9}
  \,&& \mathcal{D}(U \times V) & \to    \,&& \mathcal{D}(V) && \quad \text{ and } \quad &&   \,& \mathcal{D}(U \times V) && \to    \,&& \mathcal{D}(U) \\
    && f                   \   & \mapsto\,&& \langle S, f^{\bullet} \rangle    &&       &&     & f                   \   && \mapsto\,&& \langle T, f_{\bullet} \rangle \\
\end{alignat}</math>

Moreover, if either <math>S</math> (resp. <math>T</math>) has compact support then it also induces a continuous linear map of <math>C^\infty(U \times V) \to C^\infty(V)</math> (resp. {{nowrap|<math>C^\infty(U \times V) \to C^\infty(U)</math>).}}{{sfn|Trèves|2006|pp=416-419}}

{{Math theorem|name={{visible anchor|Fubini's theorem for distributions|text=[[Fubini's theorem]] for distributions}}{{sfn|Trèves|2006|pp=416-419}}|math_statement=
Let <math>S \in \mathcal{D}'(U)</math> and <math>T \in \mathcal{D}'(V).</math> If <math>f \in \mathcal{D}(U \times V)</math> then 
<math display=block>\langle S, \langle T, f_{\bullet} \rangle \rangle = \langle T, \langle S, f^{\bullet} \rangle \rangle.</math>
}}

{{em|The [[Tensor product|'''{{visible anchor|tensor product}}''']] of <math>S \in \mathcal{D}'(U)</math> and <math>T \in \mathcal{D}'(V),</math>}} denoted by <math>S \otimes T</math> or <math>T \otimes S,</math> is the distribution in <math>U \times V</math> defined by:{{sfn|Trèves|2006|pp=416-419}}
<math display=block>(S \otimes T)(f) := \langle S, \langle T, f_{\bullet} \rangle \rangle = \langle T, \langle S, f^{\bullet}\rangle \rangle.</math>

==Spaces of distributions==

{{See also|Spaces of test functions and distributions}}

For all <math>0 < k < \infty</math> and all <math>1 < p < \infty,</math> every one of the following canonical injections is continuous and has an [[Image of a function|image (also called the range)]] that is a [[Dense set|dense subset]] of its codomain:
<math display=block>\begin{matrix}
C_c^\infty(U) & \to & C_c^k(U) & \to & C_c^0(U) & \to & L_c^\infty(U) & \to & L_c^p(U) & \to & L_c^1(U) \\
\downarrow & &\downarrow && \downarrow \\
C^\infty(U) & \to & C^k(U) & \to & C^0(U) \\{}
\end{matrix}</math>
where the topologies on <math>L_c^q(U)</math> (<math>1 \leq q \leq \infty</math>) are defined as direct limits of the spaces <math>L_c^q(K)</math> in a manner analogous to how the topologies on <math>C_c^k(U)</math> 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 its codomain.{{sfn|Trèves|2006|pp=150-160}}

Suppose that <math>X</math> is one of the spaces <math>C_c^k(U)</math> (for <math>k \in \{0, 1, \ldots, \infty\}</math>) or <math>L^p_c(U)</math> (for <math>1 \leq p \leq \infty</math>) or <math>L^p(U)</math> (for <math>1 \leq p < \infty</math>). Because the canonical injection <math>\operatorname{In}_X : C_c^\infty(U) \to X</math> is a continuous injection whose image is dense in the codomain, this map's [[Transpose of a linear map|transpose]] <math>{}^{t}\operatorname{In}_X : X'_b \to \mathcal{D}'(U) = \left(C_c^\infty(U)\right)'_b</math> is a continuous injection. This injective transpose map thus allows the [[continuous dual space]] <math>X'</math> of <math>X</math> to be identified with a certain vector subspace of the space <math>\mathcal{D}'(U)</math> of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is {{em|not}} necessarily a [[topological embedding]].
A linear subspace of <math>\mathcal{D}'(U)</math> carrying a [[Locally convex topological vector space|locally convex]] topology that is finer than the [[subspace topology]] induced on it by <math>\mathcal{D}'(U) = \left(C_c^\infty(U)\right)'_b</math> is called '''{{em|a space of distributions}}'''.{{sfn|Trèves|2006|pp=240-252}}
Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order <math>\leq</math> some integer, distributions induced by a positive Radon measure, distributions induced by an <math>L^p</math>-function, etc.) and any representation theorem about the continuous dual space of <math>X</math> may, through the transpose <math>{}^{t}\operatorname{In}_X : X'_b \to \mathcal{D}'(U),</math> be transferred directly to elements of the space <math>\operatorname{Im} \left({}^{t}\operatorname{In}_X\right).</math>

===Radon measures===

The inclusion map <math>\operatorname{In} : C_c^\infty(U) \to C_c^0(U)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In} : (C_c^0(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection.

Note that the continuous dual space <math>(C_c^0(U))'_b</math> can be identified as the space of [[Radon measure]]s, where there is a one-to-one correspondence between the continuous linear functionals <math>T \in (C_c^0(U))'_b</math> and integral with respect to a Radon measure; that is,

* if <math>T \in (C_c^0(U))'_b</math> then there exists a Radon measure <math>\mu</math> on {{mvar|U}} such that for all <math display=inline>f \in C_c^0(U), T(f) = \int_U f \, d\mu,</math> and
* if <math>\mu</math> is a Radon measure on {{mvar|U}} then the linear functional on <math>C_c^0(U)</math> defined by sending <math display=inline>f \in C_c^0(U)</math> to <math display=inline>\int_U f \, d\mu</math> is continuous.

Through the injection <math>{}^{t}\operatorname{In} : (C_c^0(U))'_b \to \mathcal{D}'(U),</math> every Radon measure becomes a distribution on {{mvar|U}}. If <math>f</math> is a [[locally integrable]] function on {{mvar|U}} then the distribution <math display=inline>\phi \mapsto \int_U f(x) \phi(x) \, dx</math> 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 measure]]s, which shows that every Radon measure can be written as a sum of derivatives of locally <math>L^\infty</math> functions on {{mvar|U}}:

{{math theorem|name='''Theorem.'''{{sfn|Trèves|2006|pp=262–264}}|math_statement=
Suppose <math>T \in \mathcal{D}'(U)</math> is a Radon measure, where <math>U \subseteq \R^n,</math> let <math>V \subseteq U</math> be a neighborhood of the support of <math>T,</math> and let <math>I = \{p \in \N^n : |p| \leq n\}.</math> There exists a family <math>f=(f_p)_{p\in I}</math> of locally <math>L^\infty</math> functions on {{mvar|U}} such that <math>\operatorname{supp} f_p \subseteq V</math> for every <math>p\in I,</math> and
<math display=block>T = \sum_{p\in I} \partial^p f_p.</math>
Furthermore, <math>T</math> is also equal to a finite sum of derivatives of continuous functions on <math>U,</math> where each derivative has order <math>\leq 2 n.</math>
}}

====Positive Radon measures====

A linear function <math>T</math> on a space of functions is called '''{{em|positive}}''' if whenever a function <math>f</math> that belongs to the domain of <math>T</math> is non-negative (that is, <math>f</math> is real-valued and <math>f \geq 0</math>) then <math>T(f) \geq 0.</math> One may show that every positive linear functional on <math>C_c^0(U)</math> is necessarily continuous (that is, necessarily a Radon measure).{{sfn|Trèves|2006|p=218}} 
[[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 <math>f : U \to \R</math> is called '''{{em|[[locally integrable]]}}''' if it is [[Lebesgue integration|Lebesgue integrable]] over every compact subset {{mvar|K}} of {{mvar|U}}. This is a large class of functions that includes all continuous functions and all [[Lp space]] <math>L^p</math> functions. The topology on <math>\mathcal{D}(U)</math> is defined in such a fashion that any locally integrable function <math>f</math> yields a continuous linear functional on <math>\mathcal{D}(U)</math> – that is, an element of <math>\mathcal{D}'(U)</math> – denoted here by <math>T_f,</math> whose value on the test function <math>\phi</math> is given by the Lebesgue integral:
<math display=block>\langle T_f, \phi \rangle = \int_U f \phi\,dx.</math>

Conventionally, one [[Abuse of notation|abuses notation]] by identifying <math>T_f</math> with <math>f,</math> provided no confusion can arise, and thus the pairing between <math>T_f</math> and <math>\phi</math> is often written
<math display=block>\langle f, \phi \rangle = \langle T_f, \phi \rangle.</math>

If <math>f</math> and <math>g</math> are two locally integrable functions, then the associated distributions <math>T_f</math> and <math>T_g</math> are equal to the same element of <math>\mathcal{D}'(U)</math> if and only if <math>f</math> and <math>g</math> are equal [[almost everywhere]] (see, for instance, {{harvtxt|Hörmander|1983|loc=Theorem 1.2.5}}). Similarly, every [[Radon measure]] <math>\mu</math> on <math>U</math> defines an element of <math>\mathcal{D}'(U)</math> whose value on the test function <math>\phi</math> is <math display=inline>\int\phi \,d\mu.</math> As above, it is conventional to abuse notation and write the pairing between a Radon measure <math>\mu</math> and a test function <math>\phi</math> as <math>\langle \mu, \phi \rangle.</math> 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 <math>C_c^\infty(U)</math> is sequentially [[dense (topology)|dense]] in <math>\mathcal{D}'(U)</math> with respect to the strong topology on <math>\mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=300-304}} This means that for any <math>T \in \mathcal{D}'(U),</math> there is a sequence of test functions, <math>(\phi_i)_{i=1}^\infty,</math> that converges to <math>T \in \mathcal{D}'(U)</math> (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
<math display=block>\langle \phi_i, \psi \rangle \to \langle T, \psi \rangle \qquad \text{ for all } \psi \in \mathcal{D}(U).</math>

===Distributions with compact support===

The inclusion map <math>\operatorname{In}: C_c^\infty(U) \to C^\infty(U)</math> is a continuous injection whose image is dense in its codomain, so the [[Transpose of a linear map|transpose map]] <math>{}^{t}\operatorname{In}: (C^\infty(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection. Thus the image of the transpose, denoted by <math>\mathcal{E}'(U),</math> forms a space of distributions.{{sfn|Trèves|2006|pp=255-257}}

The elements of <math>\mathcal{E}'(U) = (C^\infty(U))'_b</math> can be identified as the space of distributions with compact support.{{sfn|Trèves|2006|pp=255-257}} Explicitly, if <math>T</math> is a distribution on {{mvar|U}} then the following are equivalent,

* <math>T \in \mathcal{E}'(U).</math>
* The support of <math>T</math> is compact.
* The restriction of <math>T</math> to <math>C_c^\infty(U),</math> when that space is equipped with the subspace topology inherited from <math>C^\infty(U)</math> (a coarser topology than the canonical LF topology), is continuous.{{sfn|Trèves|2006|pp=255-257}}
* There is a compact subset {{mvar|K}} of {{mvar|U}} such that for every test function <math>\phi</math> whose support is completely outside of {{mvar|K}}, we have <math>T(\phi) = 0.</math>

Compactly supported distributions define continuous linear functionals on the space <math>C^\infty(U)</math>; recall that the topology on <math>C^\infty(U)</math> is defined such that a sequence of test functions <math>\phi_k</math> converges to 0 if and only if all derivatives of <math>\phi_k</math> converge uniformly to 0 on every compact subset of {{mvar|U}}. 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 <math>C_c^\infty(U)</math> to <math>C^\infty(U).</math>

===Distributions of finite order===

Let <math>k \in \N.</math> The inclusion map <math>\operatorname{In}: C_c^\infty(U) \to C_c^k(U)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In}: (C_c^k(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection. Consequently, the image of <math>{}^{t}\operatorname{In},</math> denoted by <math>\mathcal{D}'^{k}(U),</math> forms a space of distributions. The elements of <math>\mathcal{D}'^k(U)</math> are '''{{em|the distributions of order <math>\,\leq k.</math>}}'''{{sfn|Trèves|2006|pp=258-264}} The distributions of order <math>\,\leq 0,</math> which are also called '''{{em|distributions of order {{math|0}}}}''' are exactly the distributions that are Radon measures (described above).

For <math>0 \neq k \in \N,</math> a '''{{em|distribution of order {{mvar|k}}}}''' is a distribution of order <math>\,\leq k</math> that is not a distribution of order <math>\,\leq k - 1</math>.{{sfn|Trèves|2006|pp=258-264}}

A distribution is said to be of '''{{em|finite order}}''' if there is some integer <math>k</math> such that it is a distribution of order <math>\,\leq k,</math> and the set of distributions of finite order is denoted by <math>\mathcal{D}'^{F}(U).</math> Note that if <math>k \leq l</math> then <math>\mathcal{D}'^k(U) \subseteq \mathcal{D}'^l(U)</math> so that <math>\mathcal{D}'^{F}(U) := \bigcup_{n=0}^\infty \mathcal{D}'^n(U)</math> is a vector subspace of <math>\mathcal{D}'(U)</math>, and furthermore, if and only if <math>\mathcal{D}'^{F}(U) = \mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=258-264}}

====Structure of distributions of finite order====

Every distribution with compact support in {{mvar|U}} is a distribution of finite order.{{sfn|Trèves|2006|pp=258-264}} Indeed, every distribution in {{mvar|U}} is {{em|locally}} a distribution of finite order, in the following sense:{{sfn|Trèves|2006|pp=258-264}} If {{mvar|V}} is an open and relatively compact subset of {{mvar|U}} and if <math>\rho_{VU}</math> is the restriction mapping from {{mvar|U}} to {{mvar|V}}, then the image of <math>\mathcal{D}'(U)</math> under <math>\rho_{VU}</math> is contained in <math>\mathcal{D}'^{F}(V).</math>

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 measure]]s:

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=Suppose <math>T \in \mathcal{D}'(U)</math> has finite order and <math>I =\{p \in \N^n : |p| \leq k\}.</math> Given any open subset {{mvar|V}} of {{mvar|U}} containing the support of <math>T,</math> there is a family of Radon measures in {{mvar|U}}, <math>(\mu_p)_{p \in I},</math> such that for very <math>p \in I, \operatorname{supp}(\mu_p) \subseteq V</math> and
<math display=block>T = \sum_{|p| \leq k} \partial^p \mu_p.</math>}}

'''Example.''' (Distributions of infinite order) Let <math>U := (0, \infty)</math> and for every test function <math>f,</math> let <math display=block>S f := \sum_{m=1}^\infty (\partial^m f)\left(\frac{1}{m}\right).</math>

Then <math>S</math> is a distribution of infinite order on {{mvar|U}}. Moreover, <math>S</math> can not be extended to a distribution on <math>\R</math>; that is, there exists no distribution <math>T</math> on <math>\R</math> such that the restriction of <math>T</math> to {{mvar|U}} is equal to <math>S.</math>{{sfn|Rudin|1991|pp=177-181}}

===Tempered distributions and Fourier transform {{anchor|Tempered distribution}}===

{{Redirect|Tempered distribution|tempered distributions on semisimple groups|Tempered representation}}

Defined below are the '''{{em|tempered distributions}}''', which form a subspace of <math>\mathcal{D}'(\R^n),</math> the space of distributions on <math>\R^n.</math> This is a proper subspace: while every tempered distribution is a distribution and an element of <math>\mathcal{D}'(\R^n),</math> 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 <math>\mathcal{D}'(\R^n).</math>

====Schwartz space====

The [[Schwartz space]] <math>\mathcal{S}(\R^n)</math> is the space of all smooth functions that are [[rapidly decreasing]] at infinity along with all partial derivatives. Thus <math>\phi:\R^n\to\R</math> is in the Schwartz space provided that any derivative of <math>\phi,</math> multiplied with any power of <math>|x|,</math> converges to 0 as <math>|x| \to \infty.</math> These functions form a complete TVS with a suitably defined family of [[seminorm]]s. More precisely, for any [[multi-indices]] <math>\alpha</math> and <math>\beta</math> define
<math display=block>p_{\alpha, \beta}(\phi) = \sup_{x \in \R^n} \left|x^\alpha \partial^\beta \phi(x) \right|.</math>

Then <math>\phi</math> is in the Schwartz space if all the values satisfy
<math display=block>p_{\alpha, \beta}(\phi) < \infty.</math>

The family of seminorms <math>p_{\alpha,\beta}</math> defines a [[locally convex]] topology on the Schwartz space. For <math>n = 1,</math> the seminorms are, in fact, [[Norm (mathematics)|norms]] on the Schwartz space. One can also use the following family of seminorms to define the topology:{{sfn|Trèves|2006|pp=92-94}}
<math display=block>|f|_{m,k} = \sup_{|p|\le m} \left(\sup_{x \in \R^n} \left\{(1 + |x|)^k \left|(\partial^\alpha f)(x) \right|\right\}\right), \qquad k,m \in \N.</math>

Otherwise, one can define a norm on <math>\mathcal{S}(\R^n)</math> via
<math display=block>\|\phi\|_k = \max_{|\alpha| + |\beta| \leq k} \sup_{x \in \R^n} \left| x^\alpha \partial^\beta \phi(x)\right|, \qquad k \ge 1.</math>

The Schwartz space is a [[Fréchet space]] (that is, a [[Complete topological vector space|complete]] [[Metrizable topological vector space|metrizable]] locally convex space). Because the [[Fourier transform]] changes <math>\partial^\alpha</math> into multiplication by <math>x^\alpha</math> and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

A sequence <math>\{f_i\}</math> in <math>\mathcal{S}(\R^n)</math> converges to 0 in <math>\mathcal{S}(\R^n)</math> if and only if the functions <math>(1 + |x|)^k (\partial^p f_i)(x)</math> converge to 0 uniformly in the whole of <math>\R^n,</math> which implies that such a sequence must converge to zero in <math>C^\infty(\R^n).</math>{{sfn|Trèves|2006|pp=92–94}}

<math>\mathcal{D}(\R^n)</math> is dense in <math>\mathcal{S}(\R^n).</math> The subset of all analytic Schwartz functions is dense in <math>\mathcal{S}(\R^n)</math> as well.{{sfn|Trèves|2006|p=160}}

The Schwartz space is [[Nuclear space|nuclear]], and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
<math display=block>\mathcal{S}(\R^m)\ \widehat{\otimes}\ \mathcal{S}(\R^n) \to \mathcal{S}(\R^{m+n}),</math>
where <math>\widehat{\otimes}</math> represents the completion of the [[injective tensor product]] (which in this case is identical to the completion of the [[projective tensor product]]).{{sfn|Trèves|2006|p=531}}

====Tempered distributions====

The inclusion map <math>\operatorname{In}: \mathcal{D}(\R^n) \to \mathcal{S}(\R^n)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In}: (\mathcal{S}(\R^n))'_b \to \mathcal{D}'(\R^n)</math> is also a continuous injection. Thus, the image of the transpose map, denoted by <math>\mathcal{S}'(\R^n),</math> forms a space of distributions.

The space <math>\mathcal{S}'(\R^n)</math> is called the space of {{em|tempered distributions}}. It is the [[continuous dual space]] of the Schwartz space. Equivalently, a distribution <math>T</math> is a tempered distribution if and only if
<math display=block>\left(\text{ for all } \alpha, \beta \in \N^n: \lim_{m\to \infty} p_{\alpha, \beta} (\phi_m) = 0 \right) \Longrightarrow \lim_{m\to \infty} T(\phi_m)=0.</math>

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 [[Lp space]] <math>L^p(\R^n)</math> for <math>p \geq 1</math> are tempered distributions.

The {{em|tempered distributions}} can also be characterized as {{em|slowly growing}}, meaning that each derivative of <math>T</math> grows at most as fast as some [[polynomial]]. This characterization is dual to the {{em|rapidly falling}} behaviour of the derivatives of a function in the Schwartz space, where each derivative of <math>\phi</math> decays faster than every inverse power of <math>|x|.</math> An example of a rapidly falling function is <math>|x|^n\exp (-\lambda |x|^\beta)</math> for any positive <math>n, \lambda, \beta.</math>

====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]] <math>F : \mathcal{S}(\R^n) \to \mathcal{S}(\R^n)</math> is a TVS-[[automorphism]] of the Schwartz space, and the '''{{em|Fourier transform}}''' is defined to be its [[transpose]] <math>{}^{t}F : \mathcal{S}'(\R^n) \to \mathcal{S}'(\R^n),</math> which (abusing notation) will again be denoted by <math>F.</math> So the Fourier transform of the tempered distribution <math>T</math> is defined by <math>(FT)(\psi) = T(F \psi)</math> for every Schwartz function <math>\psi.</math> <math>FT</math> 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
<math display=block>F \dfrac{dT}{dx} = ixFT</math>
and also with convolution: if <math>T</math> is a tempered distribution and <math>\psi</math> is a {{em|slowly increasing}} smooth function on <math>\R^n,</math> <math>\psi T</math> is again a tempered distribution and
<math display=block>F(\psi T) = F \psi * FT</math>
is the convolution of <math>FT</math> and <math>F \psi.</math> In particular, the Fourier transform of the constant function equal to 1 is the <math>\delta</math> distribution.

====Expressing tempered distributions as sums of derivatives====

If <math>T \in \mathcal{S}'(\R^n)</math> is a tempered distribution, then there exists a constant <math>C > 0,</math> and positive integers <math>M</math> and <math>N</math> such that for all [[Schwartz function]]s <math>\phi \in \mathcal{S}(\R^n)</math>
<math display=block>\langle T, \phi \rangle \le C\sum\nolimits_{|\alpha|\le N, |\beta|\le M}\sup_{x \in \R^n} \left|x^\alpha \partial^\beta \phi(x) \right|=C\sum\nolimits_{|\alpha|\le N, |\beta|\le M} p_{\alpha, \beta}(\phi).</math>

This estimate, along with some techniques from [[functional analysis]], can be used to show that there is a continuous slowly increasing function <math>F</math> and a multi-index <math>\alpha</math> such that
<math display=block>T = \partial^\alpha F.</math>

====Restriction of distributions to compact sets====

If <math>T \in \mathcal{D}'(\R^n),</math> then for any compact set <math>K \subseteq \R^n,</math> there exists a continuous function <math>F</math>compactly supported in <math>\R^n</math> (possibly on a larger set than {{mvar|K}} itself) and a multi-index <math>\alpha</math> such that <math>T = \partial^\alpha F</math> on <math>C_c^\infty(K).</math>

==Using holomorphic functions as test functions==

The success of the theory led to an investigation of the idea of [[hyperfunction]], in which spaces of [[holomorphic function]]s 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, [[Path integral formulation|Feynman integrals]].

==See also==

* {{annotated link|Cauchy principal value}}
* {{annotated link|Gelfand triple}}
* {{annotated link|Gelfand–Shilov space}}
* {{annotated link|Generalized function}}
* {{annotated link|Hilbert transform}}
* {{annotated link|Homogeneous distribution}}
* {{annotated link|Laplacian of the indicator}}
* {{annotated link|Limit of distributions}}
* {{annotated link|Mollifier}}
* {{annotated link|Vague topology}}

'''Differential equations related'''

* {{annotated link|Fundamental solution}}
* {{annotated link|Pseudo-differential operator}}
* {{annotated link|Weak solution}}

'''Generalizations of distributions'''

* {{annotated link|Colombeau algebra}}
* {{annotated link|Current (mathematics)}}
* {{annotated link|Distribution (number theory)}}
* {{annotated link|Distribution on a linear algebraic group}}
* {{annotated link|Green's function}}
* {{annotated link|Hyperfunction}}
* {{annotated link|Malgrange–Ehrenpreis theorem}}

==Notes==

{{reflist|group=note}}

==References==

{{reflist|29em}}

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* {{Horváth Topological Vector Spaces and Distributions Volume 1 1966}} <!-- {{sfn|Horváth|1966|p=}} -->
* {{Kolmogorov Fomin Elements of the Theory of Functions and Functional Analysis}} <!-- {{sfn|Kolmogorov|Fomin|1957|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{cite book|last=Petersen|first=Bent E.|title=Introduction to the Fourier Transform and Pseudo-Differential Operators|publisher=Pitman Publishing|publication-place=Boston, MA|year=1983}}.
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|1999|p=}} -->
* {{citation|last=Schwartz|first=Laurent|year=1954|author-link=Laurent Schwartz|title=Sur l'impossibilité de la multiplications des distributions|journal=C. R. Acad. Sci. Paris|volume=239|pages=847–848}}.
* {{citation|last=Schwartz|first=Laurent|author-link=Laurent Schwartz|title=Théorie des distributions|volume=1–2|publisher=Hermann|year=1951}}.
* {{citation|last=Sobolev|first=S.L.|author-link=Sergei Sobolev|title=Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales|journal=Mat. Sbornik|volume=1|year=1936|pages=39–72|url=http://mi.mathnet.ru/msb5358}}.
* {{citation|last1=Stein|first1=Elias|author-link1=Elias Stein|last2=Weiss|first2=Guido|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=0-691-08078-X|url-access=registration|url=https://archive.org/details/introductiontofo0000stei}}.
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* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{cite book|last=Woodward|first=P.M.|author-link=Philip Woodward|title=Probability and Information Theory with Applications to Radar|publisher=Pergamon Press|publication-place=Oxford, UK|year=1953}}

==Further reading==

* M. J. Lighthill (1959). ''Introduction to Fourier Analysis and Generalised Functions''. Cambridge University Press. {{ISBN|0-521-09128-4}} (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
* [[Vasily Vladimirov|V.S. Vladimirov]] (2002). ''Methods of the theory of generalized functions''. Taylor & Francis. {{ISBN|0-415-27356-0}}
* {{springer|id=G/g043810|title=Generalized function|first=V.S.|last=Vladimirov|author-link=Vasilii Sergeevich Vladimirov|year=2001}}.
* {{springer|id=G/g043840|title=Generalized functions, space of|first=V.S.|last=Vladimirov|author-link=Vasilii Sergeevich Vladimirov|year=2001}}.
* {{springer|id=G/g043820|title=Generalized function, derivative of a|first=V.S.|last=Vladimirov|author-link=Vasilii Sergeevich Vladimirov|year=2001}}.
* {{springer|id=G/g043830|title=Generalized functions, product of|first=V.S.|last=Vladimirov|author-link=Vasilii Sergeevich Vladimirov|year=2001}}.
* {{springer|id=G/g130030|title=Generalized function algebras|first=Michael|last=Oberguggenberger|year=2001}}.

{{Functional analysis}}
{{Topological vector spaces}}

[[Category:Articles containing proofs]]
[[Category:Functional analysis]]
[[Category:Generalizations of the derivative]]
[[Category:Generalized functions]]
[[Category:Smooth functions]]
[[Category:Schwartz distributions]]
[[Category:Differential equations]]
[[Category:Linear functionals]]