Distribution (mathematics)

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

A function $f$ is normally thought of as acting on the points in its domain by "sending" a point $x$ in its domain to the point $f(x).$ Instead of acting on points, distribution theory reinterprets functions such as $f$ as acting on test functions in a certain way. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions with compact support (bump functions are examples of test functions). Many "standard functions" (meaning for example a function that is typically encountered in a Calculus course), say for instance a continuous map $f:\mathbb {R} \to \mathbb {R} ,$ can be canonically reinterpreted as acting on test functions (instead of their usual interpretation as acting on points of their domain) via the action known as "integration against a test function"; explicitly, this means that $f$ "acts on" a test function $g$ by "sending" $g$ to the number $\textstyle \int _{\mathbb {R} }fg\,dx.$ This new action of $f$ is thus a complex (or real)-valued map, denoted by $D_{f},$ whose domain is the space of test functions; this map turns out to have two additional properties[note 1] that make it into what is known as a distribution on $\mathbb {R} .$ Distributions that arise from "standard functions" in this way are the prototypical examples of a distributions. But there are many distributions that do not arise in this way and these distributions are known as "generalized functions." Examples include the Dirac delta function or some distributions that arise via the action of "integration of test functions against measures." However, by using various methods it is nevertheless still possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.

In applications to physics and engineering, the space of test functions usually consists of smooth functions with compact support that are defined on some given non-empty open subset $U\subset \mathbb {R} ^{n}.$ This space of test functions is denoted by $C_{c}^{\infty }(U)$ or ${\mathcal {D}}(U)$ and a distribution on $U$ is by definition a linear functional on $C_{c}^{\infty }(U)$ that is continuous when $C_{c}^{\infty }(U)$ is given a topology called the canonical LF topology. This leads to the space of (all) distributions on $U$ , usually denoted by ${\mathcal {D}}'(U)$ (note the prime), which by definition is the space of all distributions on $U$ (that is, it is the continuous dual space of $C_{c}^{\infty }(U)$ ); it is these distributions that are the main focus of this article.

There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If $U=\mathbb {R} ^{n}$ then the use of Schwartz functions[note 2] as test functions gives rise to a certain subspace of ${\mathcal {D}}'(U)$ whose elements are called tempered distributions. These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace of the space of distributions ${\mathcal {D}}'(U)$ and is thus one example of a space of distributions; there are many other spaces of distributions.

There also exist other major classes of test functions that are not subsets of $C_{c}^{\infty }(U),$ such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support.[note 3] Use of analytic test functions lead to Sato's theory of hyperfunctions.

History

The practical use of distributions can be traced back to the use of Green functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.

Notation

The following notation will be used throughout this article:

$n$ is a fixed positive integer and $U$ is a fixed non-empty open subset of Euclidean space $\mathbb {R} ^{n}.$

$\mathbb {N} =\{0,1,2,\ldots \}$ denotes the natural numbers.

$k$ will denote a non-negative integer or $\infty .$

If $f$ is a function then $\operatorname {Dom} (f)$ will denote its domain and the support of $f,$ denoted by $\operatorname {supp} (f),$ is defined to be the closure of the set $\{x\in \operatorname {Dom} (f):f(x)\neq 0\}$ in $\operatorname {Dom} (f).$

For two functions $f,g:U\to \mathbb {C}$ , the following notation defines a canonical pairing:

{\displaystyle \langle f,g\rangle

A multi-index of size $n$ is an element in $\mathbb {N} ^{n}$ (given that $n$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be $n$ ). The length of a multi-index $\alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}$ is defined as $\alpha _{1}+\cdots +\alpha _{n}$ and denoted by $|\alpha |.$ Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index $\alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}$ :

{\begin{aligned}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{aligned}}

We also introduce a partial order of all multi-indices by

\beta \geq \alpha

if and only if

\beta _{i}\geq \alpha _{i}

for all

1\leq i\leq n.

When

\beta \geq \alpha

we define their multi-index binomial coefficient as:

{\binom {\beta }{\alpha }}:={\binom {\beta _{1}}{\alpha _{1}}}\cdots {\binom {\beta _{n}}{\alpha _{n}}}.

$\mathbb {K}$ will denote a certain non-empty collection of compact subsets of $U$ (described in detail below).

Definitions of test functions and distributions

In this section, we will formally define real-valued distributions on $U$ . With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb {R} ^{n}$ with any (paracompact) smooth manifold.

Notation: Suppose $k\in \{0,1,2,\ldots ,\infty \}.$
Let $C^{k}(U)$ denote the vector space of all $k$ -times continuously differentiable real-valued functions on $U$ .

For any compact subset $K \subseteq U$ , let $C^{k}(K)$ and $C^{k}(K;U)$ both denote the vector space of all those functions $f\in C^{k}(U)$ such that $\operatorname {supp} (f)\subseteq K.$
Note that $C^{k}(K)$ depends on both $K$ and $U$ but we will only indicate $K$ , where in particular, if $f\in C^{k}(K)$ then the domain of $f$ is $U$ rather than $K$ . We will use the notation $C^{k}(K;U)$ only when the notation $C^{k}(K)$ risks being ambiguous.

Clearly, every $C^{k}(K)$ contains the constant $0$ map, even if $K = \emptyset$ .

Let $C_{c}^{k}(U)$ denote the set of all $f\in C^{k}(U)$ such that $f\in C^{k}(K)$ for some compact subset $K$ of $U$ .
Equivalently, $C_{c}^{k}(U)$ is the set of all $f\in C^{k}(U)$ such that $f$ has compact support.

$C_{c}^{k}(U)$ is equal to the union of all $C^{k}(K)$ as $K$ ranges over $𝕂$ .

If $f$ is a real-valued function on $U$ , then $f$ is an element of $C_{c}^{k}(U)$ if and only if $f$ is a $C^{k}$ bump function. Every real-valued test function on $U$ is always also a complex-valued test function on $U.$

The graph of the bump function

(x,y)\in \mathbf {R} ^{2}\mapsto \Psi (r),

where

r=(x^{2}+y^{2})^{\frac {1}{2}}

and

\Psi (r)=e^{-{\frac {1}{1-r^{2}}}}\cdot \mathbf {1} _{\{|r|<1\}}.

This function is a test function on

\mathbb {R} ^{2}

and is an element of

C_{c}^{\infty }\left(\mathbb {R} ^{2}\right).

The support of this function is the closed unit disk in

\mathbb {R} ^{2}.

It is non-zero on the open unit disk and it is equal to

0

everywhere outside of it.

Note that for all $j,k\in \{0,1,2,\ldots ,\infty \}$ and any compact subsets $K$ and $L$ of $U$ , we have:

{\begin{aligned}C^{k}(K)&\subseteq C_{c}^{k}(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\leq k\\C_{c}^{k}(U)&\subseteq C_{c}^{j}(U)&&{\text{if }}j\leq k\\C^{k}(U)&\subseteq C^{j}(U)&&{\text{if }}j\leq k\\\end{aligned}}

Definition: Elements of $C_{c}^{\infty }(U)$ are called test functions on $U$ and $C_{c}^{\infty }(U)$ is called the space of test function on $U$ . We will use both ${\mathcal {D}}(U)$ and $C_{c}^{\infty }(U)$ to denote this space.

Distributions on $U$ are defined to be the continuous linear functionals on $C_{c}^{\infty }(U)$ when this vector space is endowed with a particular topology called the canonical LF-topology. This topology is unfortunately not easy to define but it is nevertheless still possible to characterize distributions in a way so that no mention of the canonical LF-topology is made.

Proposition: If $T$ is a linear functional on $C_{c}^{\infty }(U)$ then the $T$ is a distribution if and only if the following are equivalent conditions are satisfied:

For every compact subset $K\subseteq U$ there exist constants $C>0$ and $N\in \mathbb {N}$ such that for all $f\in C^{\infty }(K),$ [1]
$|T(f)|\leq C\sup\{|\partial ^{\alpha }f(x)|:x\in U,|\alpha |\leq N\};$
For every compact subset $K\subseteq U$ there exist constants $C_{K}>0$ and $N_{K}\in \mathbb {N}$ such that for all $f\in C_{c}^{\infty }(U)$ with support contained in $K,$ [2]
$|T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N_{K}\};$
For any compact subset $K\subseteq U$ and any sequence $\{f_{i}\}_{i=1}^{\infty }$ in $C^{\infty }(K),$ if $\{\partial ^{\alpha }f_{i}\}_{i=1}^{\infty }$ converges uniformly to zero on $K$ for all multi-indices $\alpha$ , then $T(f_{i})\to 0.$

The above characterizations can be used to determine whether or not a linear functional is a distribution, but more advanced uses of distributions and test functions (such as applications to differential equations) is limited if no topologies are placed on $C_{c}^{\infty }(U)$ and ${\mathcal {D}}(U).$ To define the space of distributions we must first define the canonical LF-topology, which in turn requires that several other topological vector spaces (TVSs) be defined first. We will first define a topology on $C^{\infty }(U),$ then assign every $C^{\infty }(K)$ the subspace topology induced on it by $C^{\infty }(U),$ and finally we define the canonical LF-topology on $C_{c}^{\infty }(U).$ We use the canonical LF-topology to define a topology on the space of distributions, which permits us to consider things such as convergence of distributions.

Choice of compact sets $𝕂$

Throughout, $𝕂$ will be any collection of compact subsets of $U$ such that (1) $U=\cup _{K\in \mathbb {K} }K,$ and (2) for any compact $K \subseteq U$ there exists some $K 2 \in 𝕂$ such that $K \subseteq K 2$ . The most common choices for $𝕂$ are:

The set of all compact subsets of $U$ , or
A set $\left\{{\overline {U_{1}}},{\overline {U_{2}}},\ldots \right\}$ where $U=\cup _{i=1}^{\infty }U_{i},$ and for all $i$ , ${\overline {U_{i}}}\subseteq U_{i+1}$ and $U i$ is a relatively compact non-empty open subset of $U$ (i.e. "relatively compact" means that the closure of $U i$ , in either $U$ or $\mathbb {R} ^{n},$ is compact).

We make $𝕂$ into a directed set by defining $K 1 \leq K 2$ if and only if $K 1 \subseteq K 2$ . Note that although the definitions of the subsequently defined topologies explicitly reference $𝕂$ , in reality they do not depend on the choice of $𝕂$ ; that is, if $𝕂 1$ and $𝕂 2$ are any two such collections of compact subsets of $U$ , then the topologies defined on $C^{k}(U)$ and $C_{c}^{k}(U)$ by using $𝕂 1$ in place of $𝕂$ are the same as those defined by using $𝕂 2$ in place of $𝕂$ .

Topology on C^k(U)

We now introduce the seminorms that will define the topology on $C^{k}(U).$ Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.

Suppose $k\in \{0,1,2,\ldots ,\infty \}$ and $K$ is an arbitrary compact subset of $U$ . Suppose $i$ an integer such that $0 \leq i \leq k$ [note 4] and $p$ is a multi-index with length $| p | \leq k$ . For $K \neq \emptyset$ , define:
${\begin{aligned}s_{p,K}(f)&:=\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\\[4pt]q_{i,K}(f)&:=\sup _{|p|\leq i}\left(\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\right)=\sup _{|p|\leq i}\left(s_{p,K}(f)\right)\\[4pt]r_{i,K}(f)&:=\sup _{\stackrel {|p|\leq i}{x_{0}\in K}}\left|\partial ^{p}f(x_{0})\right|\\[4pt]t_{i,K}(f)&:=\sup _{x_{0}\in K}\left(\sum _{|p|\leq i}\left|\partial ^{p}f(x_{0})\right|\right)\end{aligned}}$

while for $K = \emptyset$ we define all the functions above to be the constant $0$ map.

Each of the functions above are non-negative $ℝ$ -valued[note 5] seminorms on $C^{k}(U).$

Each of the following families of seminorms generates the same locally convex vector topology on $C^{k}(U)$ :

{\begin{alignedat}{4}(1)\quad &\{q_{i,K}&&:\;K\in \mathbb {K} ,\;\;&&i\in \mathbb {N} ,\;&&0\leq i\leq k\}\\(2)\quad &\{r_{i,K}&&:\;K\in \mathbb {K} ,\;\;&&i\in \mathbb {N} ,\;&&0\leq i\leq k\}\\(3)\quad &\{t_{i,K}&&:\;K\in \mathbb {K} ,\;\;&&i\in \mathbb {N} ,\;&&0\leq i\leq k\}\\(4)\quad &\{s_{p,K}&&:\;K\in \mathbb {K} ,\;\;&&p\in \mathbb {N} ^{n},\;&&|p|\leq k\}\end{alignedat}}

Assumption: We will henceforth assume that $C^{k}(U)$ is endowed with the locally convex topology defined by any (or equivalently, all) of the families of seminorms described above.

With this topology, $C^{k}(U)$ becomes a locally convex (non-normable) Fréchet space and all of the seminorms defined above are continuous on this space. All of the seminorms defined above are continuous functions on $C^{k}(U).$ Under this topology, a net $(f_{i})_{i\in I}$ in $C^{k}(U)$ converges to $f\in C^{k}(U)$ if and only if for every multi-index $p$ with $| p | < k + 1$ and every $K \in 𝕂$ , the net $(\partial ^{p}f_{i})_{i\in I}$ converges to $\partial ^{p}f$ uniformly on $K$ .[3] For any $k\in \{0,1,2,\ldots ,\infty \},$ any bounded subset of $C^{k+1}(U)$ is a relatively compact subset of $C^{k}(U).$ [4] In particular, a subset of $C^{\infty }(U)$ is bounded if and only if it is bounded in $C^{i}(U)$ for all $i\in \mathbb {N} .$ [4] The space $C^{k}(U)$ is a Montel space if and only if $k = \infty$ .[5]

The topology on $C^{\infty }(U)$ is the superior limit of the subspace topologies induced on $C^{\infty }(U)$ by the TVSs $C^{i}(U)$ as $i$ ranges over the non-negative integers.[3] A subset $W$ of $C^{\infty }(U)$ is open in this topology if and only if there exists $i\in \mathbb {N}$ such that $W$ is open when $C^{\infty }(U)$ is endowed with the subspace topology induced by $C^{i}(U).$

Metric defining the topology

If the family of compact sets $\mathbb {K} =\left\{{\overline {U_{1}}},{\overline {U_{2}}},\ldots \right\}$ satisfies $U=\cup _{i=1}^{\infty }U_{i}$ and ${\overline {U_{i}}}\subseteq U_{i+1}$ for all $i$ , then a complete translation-invariant metric on $C^{\infty }(U)$ can be obtained by taking a suitable countable Fréchet combination of any one of the above families. For example, using the seminorms $\left(r_{i,K_{i}}\right)_{i=1}^{\infty }$ results in

d(f,g):=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {r_{i,{\overline {U_{i}}}}(f-g)}{1+r_{i,{\overline {U_{i}}}}(f-g)}}=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {\sup _{|p|\leq i,x\in {\overline {U_{i}}}}\left|\partial ^{p}(f-g)(x)\right|}{\left[1+\sup _{|p|\leq i,x\in {\overline {U_{i}}}}\left|\partial ^{p}(f-g)(x)\right|\right]}}.

Often, it is easier to just consider seminorms.

Topology on C^k(K)

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

Assumption: For any compact subset $K \subseteq U$ , we will henceforth assume that $C^{k}(K)$ is endowed with the subspace topology it inherits from the Fréchet space $C^{k}(U).$

For any compact subset $K \subseteq U$ , $C^{k}(K)$ is a closed subspace of the Fréchet space $C^{k}(U)$ and is thus also a Fréchet space. For all compact $K, L \subseteq U$ with $K \subseteq L$ , denote the natural inclusion by $\operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L).$ Then this map is a linear embedding of TVSs (i.e. a linear map that is also a topological embedding) whose range is closed in its codomain; said differently, the topology on $C^{k}(K)$ is identical to the subspace topology it inherits from $C^{k}(L),$ and also $C^{k}(K)$ is a closed subset of $C^{k}(L).$ The interior of $C^{\infty }(K)$ relative to $C^{\infty }(U)$ is empty.[6]

If $k$ is finite then $C^{k}(K)$ is a Banach space[7] with a topology that can be defined by the norm

r_{K}(f):=\sup _{|p|<k}\left(\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\right).

And when $k = 2$ , then $C^{k}(K)$ is even a Hilbert space.[7] The space $C^{\infty }(K)$ is a distinguished Schwartz Montel space so if $C^{\infty }(K)\neq \{0\}$ then it is not normable and thus not a Banach space (although like all other $C^{k}(K),$ it is a Fréchet space).

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

The definition of $C^{k}(K)$ depends on $U$ so we will let $C^{k}(K;U)$ denote the topological space $C^{k}(K),$ which by definition is a topological subspace of $C^{k}(U).$ Suppose $V$ is an open subset of $\mathbb {R} ^{n}$ containing $U.$ Given $f\in C_{c}^{k}(U),$ its trivial extension to $V$ is by definition, the function $F:V\to \mathbb {C}$ defined by:

F(x)={\begin{cases}f(x)&x\in U,\\0&{\text{otherwise}},\end{cases}}

so that $F\in C^{k}(V).$ Let $I:C_{c}^{k}(U)\to C^{k}(V)$ denote the map that sends a function in $C_{c}^{k}(U)$ to its trivial extension on $V$ . This map is a linear injection and for every compact subset $K\subseteq U,$ we have $I\left(C^{k}(K;U)\right)=C^{k}(K;V),$ where $C^{k}(K;V)$ is the vector subspace of $C^{k}(V)$ consisting of maps with support contained in $K$ (since $K \subseteq U \subseteq V$ , $K$ is a compact subset of $V$ as well). It follows that $I\left(C_{c}^{k}(U)\right)\subseteq C_{c}^{k}(V).$ If $I$ is restricted to $C^{k}(K;U)$ then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism):

C^{k}(K;U)\to C^{k}(K;V)

and thus the next two maps (which like the previous map are defined by $f\mapsto I(f)$ ) are topological embeddings:

C^{k}(K;U)\to C^{k}(V),

C^{k}(K;U)\to C_{c}^{k}(V),

(the topology on $C_{c}^{k}(V)$ is the canonical LF topology, which is defined later). Using $C_{c}^{k}(U)\ni f\mapsto I(f)\in C_{c}^{k}(V)$ we identify $C_{c}^{k}(U)$ with its image in $C_{c}^{k}(V)\subseteq C^{k}(V).$ Because $C^{k}(K;U)\subseteq C_{c}^{k}(U),$ through this identification, $C^{k}(K;U)$ can also be considered as a subset of $C^{k}(V).$ Importantly, the subspace topology $C^{k}(K;U)$ inherits from $C^{k}(U)$ (when it is viewed as a subset of $C^{k}(U)$ ) is identical to the subspace topology that it inherits from $C^{k}(V)$ (when $C^{k}(K;U)$ is viewed instead as a subset of $C^{k}(V)$ via the identification). Thus the topology on $C^{k}(K;U)$ is independent of the open subset $U$ of $\mathbb {R} ^{n}$ that contains $K$ .[6] This justifies the practice of written $C^{k}(K)$ instead of $C^{k}(K;U).$

Topology on the spaces of test functions and distributions

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

$C_{c}^{\infty }(U)$ is called the space of test functions on $U$ and it may also be denoted by ${\mathcal {D}}(U).$

Canonical LF topology

We now define the canonical LF topology as a direct limit. It is also possible to define this topology in terms of its neighborhoods of the origin, which is described afterwards.

For any two sets $K$ and $L$ , we declare that $K \leq L$ if and only if $K \subseteq L$ , which in particular makes the collection $𝕂$ of compact subsets of $U$ into a directed set (we say that such a collection is directed by subset inclusion). For all compact $K, L \subseteq U$ with $K \subseteq L$ , there are natural inclusions

\operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L)\quad {\text{and}}\quad \operatorname {In} _{K}^{U}:C^{k}(K)\to C_{c}^{k}(U).

Recall from above that the map $\operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L)$ is a topological embedding. The collection of maps

\left\{\operatorname {In} _{K}^{L}\;:\;K,L\in \mathbb {K} \;{\text{ and }}\;K\subseteq L\right\}

forms a direct system in the category of locally convex topological vector spaces that is directed by $𝕂$ (under subset inclusion). This system's direct limit (in the category of locally convex TVSs) is the pair $(C_{c}^{k}(U),\operatorname {In} _{\bullet }^{U})$ where $\operatorname {In} _{\bullet }^{U}:=\left(\operatorname {In} _{K}^{U}\right)_{K\in \mathbb {K} }$ are the natural inclusions and where $C_{c}^{k}(U)$ is now endowed with the (unique) strongest locally convex topology making all of the inclusion maps $\operatorname {In} _{\bullet }^{U}=\left(\operatorname {In} _{K}^{U}\right)_{K\in \mathbb {K} }$ continuous.

The canonical LF topology on $C_{c}^{k}(U)$ is the finest locally convex topology on $C_{c}^{k}(U)$ making all of the inclusion maps $\operatorname {In} _{K}^{U}:C^{k}(K)\to C_{c}^{k}(U)$ continuous (where $K$ ranges over $𝕂$ ).

Assumption: As is common in mathematics literature, this article will henceforth assume that $C_{c}^{k}(U)$ is endowed with its canonical LF topology (unless explicitly stated otherwise).

Neighborhoods of the origin

If $U$ is a convex subset of $C_{c}^{k}(U),$ then $U$ is a neighborhood of the origin in the canonical LF topology if and only if it satisfies the following condition:

For all

K \in 𝕂,

U\cap C^{k}(K)

is a neighborhood of the origin in

C^{k}(K).

(CN)

Note that any convex set satisfying this condition is necessarily absorbing in $C_{c}^{k}(U).$ Since the topology of any topological vector space is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually define the canonical LF topology by declaring that a convex balanced subset $U$ is a neighborhood of the origin if and only if it satisfies condition CN.

Topology defined via differential operators

A linear differential operator in $U$ with smooth coefficients is a sum

P:=\sum _{\alpha \in \mathbb {N} ^{n}}c_{\alpha }\partial ^{\alpha }

where $c_{\alpha }\in C^{\infty }(U)$ and all but finitely many of $c_{\alpha }$ are identically $0$ . The integer $\sup\{|\alpha |:c_{\alpha }\neq 0\}$ is called the order of the differential operator $P.$ If $P$ is a linear differential operator of order $k$ then it induces a canonical linear map $C^{k}(U)\to C^{0}(U)$ defined by $\phi \mapsto P\phi ,$ where we shall reuse notation and also denote this map by $P.$ [8]

For any $1 \leq k \leq \infty$ , the canonical LF topology on $C_{c}^{k}(U)$ is the weakest locally convex TVS topology making all linear differential operators in $U$ of order $< k + 1$ into continuous maps from $C_{c}^{k}(U)$ into $C_{c}^{0}(U).$ [8]

Basic properties

Canonical LF topology's independence from $𝕂$

One benefit of defining the canonical LF topology as the direct limit of a direct system is that we may immediately use the universal property of direct limits. Another benefit is that we can use well-known results from category theory to deduce that the canonical LF topology is actually independent of the particular choice of the directed collection $𝕂$ of compact sets. And by considering different collections $𝕂$ (in particular, those $𝕂$ mentioned at the beginning of this article), we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes $C_{c}^{k}(U)$ into a Hausdorff locally convex strict LF-space (and also a strict LB-space if $k \neq \infty$ ), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).[note 6]

Universal property

From the universal property of direct limits, we know that if $u:C_{c}^{k}(U)\to Y$ is a linear map into a locally convex space $Y$ (not necessarily Hausdorff), then $u$ is continuous if and only if $u$ is bounded if and only if for every $K \in 𝕂$ , the restriction of $u$ to $C^{k}(K)$ is continuous (or bounded).[9][10]

Dependence of the canonical LF topology on $U$

Suppose $V$ is an open subset of $\mathbb {R} ^{n}$ containing $U.$ Let $I:C_{c}^{k}(U)\to C_{c}^{k}(V)$ denote the map that sends a function in $C_{c}^{k}(U)$ to its trivial extension on $V$ (which was defined above). This map is a continuous linear map.[11] If (and only if) $U \neq V$ then $I(C_{c}^{\infty }(U))$ is not a dense subset of $C_{c}^{\infty }(V)$ and $I:C_{c}^{\infty }(U)\to C_{c}^{\infty }(V)$ is not a topological embedding.[11] Consequently, if $U \neq V$ then the transpose of $I:C_{c}^{\infty }(U)\to C_{c}^{\infty }(V)$ is neither one-to-one nor onto.[11]

Bounded subsets

A subset $B$ of $C_{c}^{k}(U)$ is bounded in $C_{c}^{k}(U)$ if and only if there exists some $K \in 𝕂$ such that $B\subseteq C^{k}(K)$ and $B$ is a bounded subset of $C^{k}(K).$ [10] Moreover, if $K \subseteq U$ is compact and $S\subseteq C^{k}(K)$ then $S$ is bounded in $C^{k}(K)$ if and only if it is bounded in $C^{k}(U).$ For any $0 \leq k \leq \infty$ , any bounded subset of $C_{c}^{k+1}(U)$ (resp. $C^{k+1}(U)$ ) is a relatively compact subset of $C_{c}^{k}(U)$ (resp. $C^{k}(U)$ ), where $\infty + 1 = \infty$ .[10]

Non-metrizability

For all compact $K \subseteq U$ , the interior of $C^{k}(K)$ in $C_{c}^{k}(U)$ is empty so that $C_{c}^{k}(U)$ is of the first category in itself. It follows from Baire's theorem that $C_{c}^{k}(U)$ is not metrizable and thus also not normable (see this footnote[note 7] for an explanation of how the non-metrizable space $C_{c}^{k}(U)$ can be complete even thought it does not admit a metric). The fact that $C_{c}^{\infty }(U)$ is a nuclear Montel space makes up for the non-metrizability of $C_{c}^{\infty }(U)$ (see this footnote for a more detailed explanation).[note 8]

Relationships between spaces

Using the universal property of direct limits and the fact that the natural inclusions $\operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L)$ are all topological embedding, one may show that all of the maps $\operatorname {In} _{K}^{U}:C^{k}(K)\to C_{c}^{k}(U)$ are also topological embeddings. Said differently, the topology on $C^{k}(K)$ is identical to the subspace topology that it inherits from $C_{c}^{k}(U),$ where recall that $C^{k}(K)$ 's topology was defined to be the subspace topology induced on it by $C^{k}(U).$ In particular, both $C_{c}^{k}(U)$ and $C^{k}(U)$ induces the same subspace topology on $C^{k}(K).$ However, this does not imply that the canonical LF topology on $C_{c}^{k}(U)$ is equal to the subspace topology induced on $C_{c}^{k}(U)$ by $C^{k}(U)$ ; these two topologies on $C_{c}^{k}(U)$ are in fact never equal to each other since the canonical LF topology is never metrizable while the subspace topology induced on it by $C^{k}(U)$ is metrizable (since recall that $C^{k}(U)$ is metrizable). The canonical LF topology on $C_{c}^{k}(U)$ is actually strictly finer than the subspace topology that it inherits from $C^{k}(U)$ (thus the natural inclusion $C_{c}^{k}(U)\to C^{k}(U)$ is continuous but not a topological embedding).[7]

Indeed, the canonical LF topology is so fine that if $C_{c}^{\infty }(U)\to X$ denotes some linear map that is a "natural inclusion" (such as $C_{c}^{\infty }(U)\to C^{k}(U),$ or $C_{c}^{\infty }(U)\to L^{p}(U),$ or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions, Radon measures, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on $C_{c}^{\infty }(U),$ the fine nature of the canonical LF topology means that more linear functionals on $C_{c}^{\infty }(U)$ end up being continuous ("more" means as compared to a coarser topology that we could have placed on $C_{c}^{\infty }(U)$ such as for instance, the subspace topology induced by some $C^{k}(U),$ which although it would have made $C_{c}^{\infty }(U)$ metrizable, it would have also resulted in fewer linear functionals on $C_{c}^{\infty }(U)$ being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making $C_{c}^{\infty }(U)$ into a complete TVS[12]).

Other properties

The differentiation map $C_{c}^{\infty }(U)\to C_{c}^{\infty }(U)$ is a surjective continuous linear operator.[13]
The bilinear multiplication map $C^{\infty }(\mathbb {R} ^{m})\times C_{c}^{\infty }(\mathbb {R} ^{n})\to C_{c}^{\infty }(\mathbb {R} ^{m+n})$ given by $(f,g)\mapsto fg$ is not continuous; it is however, hypocontinuous.[14]

Distributions

As discussed earlier, continuous linear functionals on a $C_{c}^{\infty }(U)$ are known as distributions on $U$ . Thus the set of all distributions on $U$ is the continuous dual space of $C_{c}^{\infty }(U),$ which when endowed with the strong dual topology is denoted by ${\mathcal {D}}'(U).$

By definition, a distribution on $U$ is defined to be a continuous linear functional on $C_{c}^{\infty }(U).$ Said differently, a distribution on $U$ is an element of the continuous dual space of $C_{c}^{\infty }(U)$ when $C_{c}^{\infty }(U)$ is endowed with its canonical LF topology.

We have the canonical duality pairing between a distribution $T$ on $U$ and a test function $f\in C_{c}^{\infty }(U),$ which is denoted using angle brackets by

{\displaystyle {\begin{cases}{\mathcal {D}}'(U)\times C_{c}^{\infty }(U)\to \mathbb {R} \\(T,f)\mapsto \langle T,f\rangle

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

Characterizations of distributions

Proposition. If $T$ is a linear functional on $C_{c}^{\infty }(U)$ then the following are equivalent:

$T$ is a distribution;
(definition) $T$ is continuous;
$T$ is continuous at the origin;
$T$ is uniformly continuous;
$T$ is a bounded operator;
T is sequentially continuous;
- explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to some $f\in C_{c}^{\infty }(U),$ $\lim _{i\to \infty }T\left(f_{i}\right)=T(f);$ [note 9]
T is sequentially continuous at the origin; in other words, T maps null sequences[note 10] to null sequences;
- explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to the origin (such a sequence is called a null sequence), $\lim _{i\to \infty }T\left(f_{i}\right)=0;$
- a null sequence is by definition a sequence that converges to the origin;
T maps null sequences to bounded subsets;
- explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to the origin, the sequence $\left(T\left(f_{i}\right)\right)_{i=1}^{\infty }$ is bounded;
T maps Mackey convergence null sequences[note 11] to bounded subsets;
- explicitly, for every Mackey convergent null sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U),$ the sequence $\left(T\left(f_{i}\right)\right)_{i=1}^{\infty }$ is bounded;
- a sequence $f • = (f i) \infty i =1$ is said to be Mackey convergent to $0$ if there exists a divergent sequence $r • = (r i) \infty i =1 \to \infty$ of positive real number such that the sequence $(r i f i) \infty i =1$ is bounded; every sequence that is Mackey convergent to $0$ necessarily converges to the origin (in the usual sense);
The kernel of $T$ is a closed subspace of $C_{c}^{\infty }(U);$
The graph of $T$ is a closed;
There exists a continuous seminorm $g$ on $C_{c}^{\infty }(U)$ such that $|T|\leq g;$
There exists a constant $C > 0$ , a collection of continuous seminorms, ${\mathcal {P}},$ that defines the canonical LF topology of $C_{c}^{\infty }(U),$ and a finite subset $\{g_{1},\ldots ,g_{m}\}\subseteq {\mathcal {P}}$ such that $|T|\leq C(g_{1}+\cdots g_{m});$ [note 12]
For every compact subset $K\subseteq U$ there exist constants $C>0$ and $N\in \mathbb {N}$ such that for all $f\in C^{\infty }(K),$ [1]
$|T(f)|\leq C\sup\{|\partial ^{p}f(x)|:x\in U,|\alpha |\leq N\};$
For every compact subset $K\subseteq U$ there exist constants $C_{K}>0$ and $N_{K}\in \mathbb {N}$ such that for all $f\in C_{c}^{\infty }(U)$ with support contained in $K,$ [15]
$|T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N_{K}\};$
For any compact subset $K\subseteq U$ and any sequence $\{f_{i}\}_{i=1}^{\infty }$ in $C^{\infty }(K),$ if $\{\partial ^{p}f_{i}\}_{i=1}^{\infty }$ converges uniformly to zero for all multi-indices $p$ , then $T(f_{i})\to 0;$
Any of the three statements immediately above (i.e. statements 14, 15, and 16) but with the additional requirement that compact set $K$ belongs to $𝕂$ .

Topology on the space of distributions

Definition and notation: The space of distributions on $U$ , denoted by ${\mathcal {D}}'(U),$ is the continuous dual space of $C_{c}^{\infty }(U)$ endowed with the topology of uniform convergence on bounded subsets of $C_{c}^{\infty }(U).$ [7] More succinctly, the space of distributions on $U$ is ${\mathcal {D}}'(U):=(C_{c}^{\infty }(U))'_{b}.$

The topology of uniform convergence on bounded subsets is also called the strong dual topology.[note 13] This topology is chosen because it is with this topology that ${\mathcal {D}}'(U)$ becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.[16] No matter what dual topology is placed on ${\mathcal {D}}'(U)$ ,[note 14] a sequence of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net). No matter which topology is chosen, ${\mathcal {D}}'(U)$ will be a non-metrizable, locally convex topological vector space. The space ${\mathcal {D}}'(U)$ is separable[17] and has the strong Pytkeev property[18] but it is neither a k-space[18] nor a sequential space,[17] which in particular implies that it is not metrizable and also that its topology can not be defined using only sequences.

Topological properties

Topological vector space categories

The canonical LF topology makes $C_{c}^{k}(U)$ into a complete distinguished strict LF-space (and a strict LB-space if and only if $k \neq \infty$ [19]), which implies that $C_{c}^{k}(U)$ is a meager subset of itself.[20] Furthermore, $C_{c}^{k}(U),$ as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of $C_{c}^{k}(U)$ is a Fréchet space if and only if $k \neq \infty$ so in particular, the strong dual of $C_{c}^{\infty }(U),$ which is the space ${\mathcal {D}}'(U)$ of distributions on $U$ , is not metrizable (note that the weak-* topology on ${\mathcal {D}}'(U)$ also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives ${\mathcal {D}}'(U)$ ).

The three spaces $C_{c}^{\infty }(U),$ $C^{\infty }(U),$ and the Schwartz space ${\mathcal {S}}(\mathbb {R} ^{n}),$ as well as the strong duals of each of these three spaces, are complete nuclear[21] Montel[22] bornological spaces, which implies that all six of these locally convex spaces are also paracompact[23] reflexive barrelled Mackey spaces. The spaces $C^{\infty }(U)$ and ${\mathcal {S}}(\mathbb {R} ^{n})$ are both distinguished Fréchet spaces. Moreover, both $C_{c}^{\infty }(U)$ and ${\mathcal {S}}(\mathbb {R} ^{n})$ are Schwartz TVSs.

Convergent sequences

Convergent sequences and their insufficiency to describe topologies

The strong dual spaces of $C^{\infty }(U)$ and ${\mathcal {S}}(\mathbb {R} ^{n})$ are sequential spaces but not Fréchet-Urysohn spaces.[17] Moreover, neither the space of test functions $C_{c}^{\infty }(U)$ nor its strong dual ${\mathcal {D}}'(U)$ is a sequential space (not even an Ascoli space),[17][24] which in particular implies that their topologies can not be defined entirely in terms of convergent sequences.

A sequence $(f_{i})_{i=1}^{\infty }$ in $C_{c}^{k}(U)$ converges in $C_{c}^{k}(U)$ if and only if there exists some $K \in 𝕂$ such that $C^{k}(K)$ contains this sequence and this sequence converges in $C^{k}(K)$ ; equivalently, it converges if and only if the following two conditions hold:[25]

There is a compact set $K \subseteq U$ containing the supports of all $f_{i}.$
For each multi-index $α$ , the sequence of partial derivatives $\partial ^{\alpha }f_{i}$ tends uniformly to $\partial ^{\alpha }f.$

Neither the space $C_{c}^{\infty }(U)$ nor its strong dual ${\mathcal {D}}'(U)$ is a sequential space,[17][24] and consequently, their topologies can not be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is not enough to define the canonical LF topology on $C_{c}^{\infty }(U).$ The same can be said of the strong dual topology on ${\mathcal {D}}'(U).$

What sequences do characterize

Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology,[26] which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually define the convergence of a sequence of distributions; this is fine for sequences but it does not extend to the convergence of nets of distributions since a net may converge pointwise but fail to convergence in the strong dual topology).

Sequences characterize continuity of linear maps valued in locally convex space. Suppose $X$ is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map $F : X \to Y$ into a locally convex space $Y$ is continuous if and only if it maps null sequences[note 10] in $X$ to bounded subsets of $Y$ .[note 15] More generally, such a linear map $F : X \to Y$ is continuous if and only if it maps Mackey convergent null sequences[note 11] to bounded subsets of $Y.$ So in particular, if a linear map $F : X \to Y$ into a locally convex space is sequentially continuous at the origin then it is continuous.[27] However, this does not necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs.

For every $k\in \{0,1,\ldots ,\infty \},C_{c}^{\infty }(U)$ is sequentially dense in $C_{c}^{k}(U).$ [28] Furthermore, $\{D_{\phi }:\phi \in C_{c}^{\infty }(U)\}$ is a sequentially dense subset of ${\mathcal {D}}'(U)$ (with its strong dual topology)[29] and also a sequentially dense subset of the strong dual space of $C^{\infty }(U).$ [29]

Sequences of distributions

A sequence of distributions $(T_{i})_{i=1}^{\infty }$ converges with respect to the weak-* topology on ${\mathcal {D}}'(U)$ to a distribution $T$ if and only if

\langle T_{i},f\rangle \to \langle T,f\rangle

for every test function $f\in {\mathcal {D}}(U).$ For example, if $f_{m}:\mathbb {R} \to \mathbb {R}$ is the function

f_{m}(x)={\begin{cases}m&{\text{if }}x\in [0,{\frac {1}{m}}]\\0&{\text{otherwise}}\end{cases}}

and $T m$ is the distribution corresponding to $f_{m},$ then

\langle T_{m},f\rangle =m\int _{0}^{\frac {1}{m}}f(x)\,dx\to f(0)=\langle \delta ,f\rangle

as $m \to \infty$ , so $T m \to δ$ in ${\mathcal {D}}'(\mathbb {R} ).$ Thus, for large $m$ , the function $f_{m}$ can be regarded as an approximation of the Dirac delta distribution.

Other properties

The strong dual space of ${\mathcal {D}}'(U)$ is TVS isomorphic to $C_{c}^{\infty }(U)$ via the canonical TVS-isomorphism $C_{c}^{\infty }(U)\to ({\mathcal {D}}'(U))'_{b}$ defined by sending $f\in C_{c}^{\infty }(U)$ to value at $f$ (i.e. to the linear functional on ${\mathcal {D}}'(U)$ defined by sending $d\in {\mathcal {D}}'(U)$ to $d(f)$ );
On any bounded subset of ${\mathcal {D}}'(U),$ the weak and strong subspace topologies coincide; the same is true for $C_{c}^{\infty }(U)$ ;
Every weakly convergent sequence in ${\mathcal {D}}'(U)$ is strongly convergent (although this does not extend to nets).

Localization of distributions

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

Restrictions to an open subset

Let $U$ and $V$ be open subsets of $\mathbb {R} ^{n}$ with $V \subseteq U$ . Let $E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U)$ be the operator which extends by zero a given smooth function compactly supported in $V$ to a smooth function compactly supported in the larger set $U$ . The transpose of $E_{VU}$ is called the restriction mapping and is denoted by $\rho _{VU}:={}^{t}E_{VU}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(V).$

The map $E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U)$ is a continuous injection where if $V \subseteq U$ then it is not a topological embedding and its range is not dense in ${\mathcal {D}}(U),$ which implies that this map's transpose is neither injective nor surjective and that the topology that $E_{VU}$ transfers from ${\mathcal {D}}(V)$ onto its image is strictly finer than the subspace topology that ${\mathcal {D}}(U)$ induces on this same set.[11] A distribution $S\in {\mathcal {D}}'(V)$ is said to be extendible to $U$ if it belongs to the range of the transpose of $E_{VU}$ and it is called extendible if it is extendable to $\mathbb {R} ^{n}.$ [11]

For any distribution $T\in {\mathcal {D}}'(U),$ the restriction $ρ VU (T)$ is a distribution in ${\mathcal {D}}'(V)$ defined by:

\qquad \langle \rho _{VU}T,\phi \rangle =\langle T,E_{VU}\phi \rangle \quad {\text{ for all }}\phi \in {\mathcal {D}}(V).

Unless $U = V$ , the restriction to $V$ is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of $V$ . For instance, if $U = ℝ$ and $V = (0, 2)$ , then the distribution

T(x)=\sum _{n=1}^{\infty }n\,\delta \left(x-{\frac {1}{n}}\right)

is in ${\mathcal {D}}'(V)$ but admits no extension to ${\mathcal {D}}'(U).$

Gluing and distributions that vanish in a set

Theorem[30] — Let $(U_{i})_{i\in I}$ be a collection of open subsets of $\mathbb {R} ^{n}.$ For each $i\in I,$ let $T_{i}\in {\mathcal {D}}'(U_{i})$ and suppose that for all $i,j\in I,$ the restriction of $T_{i}$ to $U_{i}\cap U_{j}$ is equal to the restriction of $T_{j}$ to $U_{i}\cap U_{j}$ (note that both restrictions are elements of ${\mathcal {D}}'(U_{i}\cap U_{j})$ ). Then there exists a unique $T\in {\mathcal {D}}'(\cup _{i\in I}U_{i})$ such that for all $i\in I,$ the restriction of $T$ to $U_{i}$ is equal to $T_{i}.$

Let $V$ be an open subset of $U$ . $T\in {\mathcal {D}}'(U)$ is said to vanish in $V$ if for all $f\in {\mathcal {D}}(U)$ such that $\operatorname {supp} (f)\subseteq V$ we have $Tf=0.$ $T$ vanishes in $V$ if and only if the restriction of $T$ to $V$ is equal to 0, or equivalently, if and only if $T$ lies in the kernel of the restriction map $ρ VU$ .

Corollary.[30] Let

(U_{i})_{i\in I}

be a collection of open subsets of

\mathbb {R} ^{n}

and let

T\in {\mathcal {D}}'(\cup _{i\in I}U_{i}).

T = 0

if and only if for each

i\in I,

the restriction of

T

to

U_{i}

is equal to 0.

Corollary.[30] The union of all open subsets of

U

in which a distribution

T

vanishes is an open subset of

U

in which

T

vanishes.

Support of a distribution

This last corollary implies that for every distribution $T$ on $U$ , there exists a unique largest subset $V$ of $U$ such that $T$ vanishes in $V$ (and does not vanish in any open subset of $U$ that is not contained in $V$ ); the complement in $U$ of this unique largest open subset is called the support of $T$ .[30] Thus

\operatorname {supp} (T)=U\setminus \bigcup \{V\mid \rho _{VU}T=0\}.

If $f$ is a locally integrable function on $U$ and if $D_{f}$ is its associated distribution, then the support of $D_{f}$ is the smallest closed subset of $U$ in the complement of which $f$ is almost everywhere equal to 0.[30] If $f$ is continuous, then the support of $D_{f}$ is equal to the closure of the set of points in $U$ at which $f$ does not vanish.[30] The support of the distribution associated with the Dirac measure at a point $x_{0}$ is the set $\{x_{0}\}.$ [30] If the support of a test function $f$ does not intersect the support of a distribution $T$ then $Tf = 0$ . A distribution $T$ is 0 if and only if its support is empty. If $f\in C^{\infty }(U)$ is identically 1 on some open set containing the support of a distribution $T$ then $fT = T$ . If the support of a distribution $T$ is compact then it has finite order and furthermore, there is a constant C and a non-negative integer N such that:[6]

\qquad |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).

If $T$ has compact support then it has a unique extension to a continuous linear functional ${\widehat {T}}$ on $C^{\infty }(U)$ ; this functional can be defined by ${\widehat {T}}(f):=T(\psi f),$ where $\psi \in {\mathcal {D}}(U)$ is any function that is identically 1 on an open set containing the support of $T$ .[6]

If $S,T\in {\mathcal {D}}'(U)$ and $\lambda \neq 0$ then $\operatorname {supp} (S+T)\subseteq \operatorname {supp} (S)\cup \operatorname {supp} (T)$ and $\operatorname {supp} (\lambda T)=\operatorname {supp} (T).$ Thus, distributions with support in a given subset $A\subseteq U$ form a vector subspace of ${\mathcal {D}}'(U)$ ; such a subspace is weakly closed in ${\mathcal {D}}'(U)$ if and only if A is closed in $U$ .[31] Furthermore, if $P$ is a differential operator in $U$ , then for all distributions $T$ on $U$ and all $f\in C^{\infty }(U)$ we have $\operatorname {supp} (P(x,\partial )T)\subseteq \operatorname {supp} (T)$ and $\operatorname {supp} (fT)\subseteq \operatorname {supp} (f)\cap \operatorname {supp} (T).$ [31]

Distributions with compact support

Support in a point set and Dirac measures

For any $x\in U,$ let $\delta _{x}\in {\mathcal {D}}'(U)$ denote the distribution induced by the Dirac measure at x. For any $x_{0}\in U$ and distribution $T\in {\mathcal {D}}'(U),$ the support of $T$ is contained in $\{x_{0}\}$ if and only if $T$ is a finite linear combination of derivatives of the Dirac measure at $x_{0}.$ [32] If in addition the order of $T$ is $\leq k$ then there exist constants $\alpha _{p}$ such that:[33]

T=\sum _{|p|\leq k}\alpha _{p}\partial ^{p}\delta _{x_{0}}.

Said differently, if $T$ has support at a single point $\{P\},$ then $T$ is in fact a finite linear combination of distributional derivatives of the $δ$ function at $P$ . That is, there exists an integer $m$ and complex constants $a α$ such that

T=\sum _{|\alpha |\leq m}a_{\alpha }\partial ^{\alpha }(\tau _{P}\delta )

where $\tau _{P}$ is the translation operator.

Distribution with compact support

Theorem[6] — Suppose $T$ is a distribution on $U$ with compact support $K$ . There exists a continuous function $f$ defined on $U$ and a multi-index $p$ such that

T=\partial ^{p}f,

where the derivatives are understood in the sense of distributions. That is, for all test functions $\phi$ on $U$ ,

T\phi =(-1)^{|p|}\int _{U}f(x)(\partial ^{p}\phi )(x)\,dx.

Distributions of finite order with support in an open subset

Theorem[6] — Suppose $T$ is a distribution on $U$ with compact support $K$ and let $V$ be an open subset of $U$ containing $K$ . Since every distribution with compact support has finite order, take $N$ to be the order of $T$ and define $P:=\{0,1,\ldots ,N+2\}^{n}.$ There exists a family of continuous functions $(f_{p})_{p\in P}$ defined on $U$ with support in $V$ such that

T=\sum _{p\in P}\partial ^{p}f_{p},

where the derivatives are understood in the sense of distributions. That is, for all test functions $\phi$ on $U$ ,

T\phi =\sum _{p\in P}(-1)^{|p|}\int _{U}f_{p}(x)(\partial ^{p}\phi )(x)\,dx.

Global structure of distributions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of ${\mathcal {D}}(U)$ (or the Schwartz space ${\mathcal {S}}(\mathbb {R} ^{n})$ for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

Distributions as sheafs

Theorem[34] — Let $T$ be a distribution on $U$ . There exists a sequence $(T_{i})_{i=1}^{\infty }$ in ${\mathcal {D}}'(U)$ such that each $T i$ has compact support and every compact subset $K \subseteq U$ intersects the support of only finitely many $T i$ , and the sequence of partial sums $(S_{j})_{j=1}^{\infty },$ defined by $S_{j}:=T_{1}+\cdots +T_{j},$ converges in ${\mathcal {D}}'(U)$ to $T$ ; in other words we have:

T=\sum _{i=1}^{\infty }T_{i}.

Recall that a sequence converges in ${\mathcal {D}}'(U)$ (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 $U$ as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on $U$ . In other words for arbitrary $T\in {\mathcal {D}}'(U)$ we can write:

T=\sum _{i=1}^{\infty }\sum _{p\in P_{i}}\partial ^{p}f_{ip},

where $P_{1},P_{2},\ldots$ are finite sets of multi-indices and the functions $f_{ip}$ are continuous.

Theorem[35] — Let $T$ be a distribution on $U$ . For every multi-index $p$ there exists a continuous function $g p$ on $U$ such that

any compact subset $K$ of $U$ intersects the support of only finitely many $g p$ , and
$T=\sum \nolimits _{p}\partial ^{p}g_{p}.$

Moreover, if $T$ has finite order, then one can choose $g p$ in such a way that only finitely many of them are non-zero.

Note that the infinite sum above is well-defined as a distribution. The value of $T$ for a given $f\in {\mathcal {D}}(U)$ can be computed using the finitely many $g α$ that intersect the support of $f.$

Operations on distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ is a linear map which is continuous with respect to the weak topology, then it is possible to extend $A$ to a map $A:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ by passing to the limit.[note 16]

Preliminaries: Transpose of a linear operator

Operations on distributions and spaces of distributions are often defined by means of the transpose of a linear operator because it provides a unified approach that the many definitions in the theory of distributions and because of its many well-known topological properties.[36] In general the transpose of a continuous linear map $A:X\to Y$ is the linear map ${}^{t}A:Y'\to X'$ defined by ${}^{t}A(y'):=y'\circ A,$ or equivalently, it is the unique map satisfying $\langle y',A(x)\rangle =\left\langle {}^{t}A(y'),x\right\rangle$ for all $x\in X$ and all $y'\in Y'.$ Since $A$ is continuous, the transpose ${}^{t}A:Y'\to X'$ is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).

In the context of distributions, the characterization of the transpose can be refined slightly. Let $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ be a continuous linear map. Then by definition, the transpose of $A$ is the unique linear operator $A^{t}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ that satisfies:

\langle {}^{t}A(T),\phi \rangle =\langle T,A(\phi )\rangle

for all

\phi \in {\mathcal {D}}(U)

and all

T\in {\mathcal {D}}'(U).

However, since the image of ${\mathcal {D}}(U)$ is dense in ${\mathcal {D}}'(U),$ it is sufficient that the above equality hold for all distributions of the form $T=D_{\psi }$ where $\psi \in {\mathcal {D}}(U).$ Explicitly, this means that the above condition holds if and only if the condition below holds:

\langle {}^{t}A(D_{\psi }),\phi \rangle =\langle D_{\psi },A(\phi )\rangle =\langle \psi ,A(\phi )\rangle =\int _{U}\psi (A\phi )\,dx

for all

\phi ,\psi \in {\mathcal {D}}(U).

Differentiation of distributions

Let $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ is the partial derivative operator ${\tfrac {\partial }{\partial x_{k}}}.$ In order to extend $A$ we compute its transpose:

{\begin{aligned}\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 \end{aligned}}

Therefore ${}^{t}A=-A.$ Therefore the partial derivative of $T$ with respect to the coordinate $x_{k}$ is defined by the formula

\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).

With this definition, every distribution is infinitely differentiable, and the derivative in the direction $x_{k}$ is a linear operator on ${\mathcal {D}}'(U).$

More generally, if $\alpha$ is an arbitrary multi-index, then the partial derivative $\partial ^{\alpha }T$ of the distribution $T\in {\mathcal {D}}'(U)$ is defined by

\langle \partial ^{\alpha }T,\phi \rangle =(-1)^{|\alpha |}\langle T,\partial ^{\alpha }\phi \rangle \qquad {\text{for all }}\phi \in {\mathcal {D}}(U).

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

If $T$ is a distribution in $ℝ$ then

\lim _{x\to 0}{\frac {T-\tau _{x}T}{x}}=T'\in {\mathcal {D}}'(\mathbb {R} ),

where $T'$ is the derivative of $T$ and $τ x$ is translation by $x$ ; thus the derivative of $T$ may be viewed as a limit of quotients.[37]

Differential operators acting on smooth functions

A linear differential operator in $U$ with smooth coefficients acts on the space of smooth functions on $U.$ Given $\textstyle P:=\sum \nolimits _{\alpha }c_{\alpha }\partial ^{\alpha }$ we would like to define a continuous linear map, $D_{P}$ that extends the action of $P$ on $C^{\infty }(U)$ to distributions on $U.$ In other words we would like to define $D_{P}$ such that the following diagram commutes:

{\begin{matrix}{\mathcal {D}}'(U)&{\stackrel {D_{P}}{\longrightarrow }}&{\mathcal {D}}'(U)\\\uparrow &&\uparrow \\C^{\infty }(U)&{\stackrel {P}{\longrightarrow }}&C^{\infty }(U)\end{matrix}}

Where the vertical maps are given by assigning $f\in C^{\infty }(U)$ its canonical distribution $D_{f}\in {\mathcal {D}}'(U),$ which is defined by: $D_{f}(\phi )=\langle f,\phi \rangle$ for all $\phi \in {\mathcal {D}}(U).$ With this notation the diagram commuting is equivalent to:

D_{P(f)}=D_{P}D_{f}\qquad {\text{ for all }}f\in C^{\infty }(U).

In order to find $D_{P}$ we consider the transpose ${}^{t}P:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ of the continuous induced map $P:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ defined by $\phi \mapsto P(\phi ).$ As discussed above, for any $\phi \in {\mathcal {D}}(U),$ the transpose may be calculated by:

{\begin{aligned}\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)\,dx\end{aligned}}

For the last line we used integration by parts combined with the fact that $\phi$ and therefore all the functions $f(x)c_{\alpha }(x)\partial ^{\alpha }\phi (x)$ have compact support.[note 17] Continuing the calculation above we have for all $\phi \in {\mathcal {D}}(U):$

{\begin{aligned}\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 \leq \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 \leq \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{aligned}}

Define the formal transpose of $P,$ which will be denoted by $P_{*}$ to avoid confusion with the transpose map, to be the following differential operator on $U$ :

P_{*}:=\sum \nolimits _{\alpha }b_{\alpha }\partial ^{\alpha }

The computations above have shown that:

Lemma. Let

P

be a linear differential operator with smooth coefficients in

U.

Then for all

\phi \in {\mathcal {D}}(U)

we have

\left\langle {}^{t}P(D_{f}),\phi \right\rangle =\left\langle D_{P_{*}(f)},\phi \right\rangle ,

which is equivalent to:

{}^{t}P(D_{f})=D_{P_{*}(f)}.

The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, i.e. $P_{**}=P,$ [8] enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator $P_{*}:C_{c}^{\infty }(U)\to C_{c}^{\infty }(U)$ defined by $\phi \mapsto P_{*}(\phi ).$ We claim that the transpose of this map, ${}^{t}P_{*}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U),$ can be taken as $D_{P}.$ To see this, for every $\phi \in {\mathcal {D}}(U)$ , compute its action on a distribution of the form $D_{f}$ with $f\in C^{\infty }(U)$ :

{\begin{aligned}\left\langle {}^{t}P_{*}(D_{f}),\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{aligned}}

We call the continuous linear operator $D_{P}:={}^{t}P_{*}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ the differential operator on distributions extending $P$ .[8] Its action on an arbitrary distribution $S$ is defined via:

D_{P}(S)(\phi )=S(P_{*}(\phi ))\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).

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

Multiplication of distributions by smooth functions

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

We now give an alternative presentation of multiplication by a smooth function. If $m:U\to \mathbb {R}$ is a smooth function and $T$ is a distribution on $U$ , then the product mT is defined by

\langle mT,\phi \rangle =\langle T,m\phi \rangle \qquad {\text{for all }}\phi \in {\mathcal {D}}(U).

This definition coincides with the transpose definition since if $M:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ is the operator of multiplication by the function $m$ (i.e., $(M\phi )(x)=m(x)\phi (x)$ ), then

\int _{U}(M\phi )(x)\psi (x)\,dx=\int _{U}m(x)\phi (x)\psi (x)\,dx=\int _{U}\phi (x)m(x)\psi (x)\,dx=\int _{U}\phi (x)(M\psi )(x)\,dx,

so that ${}^{t}M=M.$

Under multiplication by smooth functions, ${\mathcal {D}}'(U)$ is a module over the ring $C^{\infty }(U).$ With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if $δ'$ is the Dirac delta distribution on $ℝ$ , then $mδ = m (0) δ$ , and if $δ'$ is the derivative of the delta distribution, then

m\delta '=m(0)\delta '-m'\delta =m(0)\delta '-m'(0)\delta .

The bilinear multiplication map $C^{\infty }(\mathbb {R} ^{n})\times {\mathcal {D}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})$ given by $(f,T)\mapsto fT$ is not continuous; it is however, hypocontinuous.[14]

Example. For any distribution $T$ , the product of $T$ with the function that is identically $1$ on $U$ is equal to $T$ .

Example. Suppose $(f_{i})_{i=1}^{\infty }$ is a sequence of test functions on $U$ that converges to the constant function $1\in C^{\infty }(U).$ For any distribution $T$ on $U$ , the sequence $(f_{i}T)_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U).$ [38]

If $(T_{i})_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U)$ and $(f_{i})_{i=1}^{\infty }$ converges to $f\in C^{\infty }(U)$ then $(f_{i}T_{i})_{i=1}^{\infty }$ converges to $fT\in {\mathcal {D}}'(U).$

Problem of multiplying distributions

It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if p.v. $.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} 1 / x$ is the distribution obtained by the Cauchy principal value

\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}}(\mathbb {R} ).

If $δ$ is the Dirac delta distribution then

(\delta \times x)\times \operatorname {p.v.} {\frac {1}{x}}=0

but

\delta \times \left(x\times \operatorname {p.v.} {\frac {1}{x}}\right)=\delta

so the product of a distribution by a smooth function (which is always well defined) cannot be extended to an associative product on the space of distributions.

Thus, nonlinear problems cannot be posed in general and thus not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) causal perturbation theory. This does not solve the problem in other situations. Many other interesting theories are non linear, like for example the Navier–Stokes equations of fluid dynamics.

Several not entirely satisfactory theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today.

Inspired by Lyons' rough path theory,[39] Martin Hairer proposed a consistent way of multiplying distributions with certain structure (regularity structures[40]), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.

Composition with a smooth function

Let $T$ be a distribution on $U.$ Let $V$ be an open set in $\mathbb {R} ^{n},$ and $F : V \to U$ . If $F$ is a submersion, it is possible to define

T\circ F\in {\mathcal {D}}'(V).

This is the composition of the distribution $T$ with $F$ , and is also called the pullback of $T$ along $F$ , sometimes written

F^{\sharp }:T\mapsto F^{\sharp }T=T\circ F.

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

The condition that $F$ be a submersion is equivalent to the requirement that the Jacobian derivative $d F (x)$ of $F$ is a surjective linear map for every $x \in V$ . A necessary (but not sufficient) condition for extending $F #$ to distributions is that $F$ be an open mapping.[41] The inverse function theorem ensures that a submersion satisfies this condition.

If $F$ is a submersion, then $F #$ is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since $F #$ is a continuous linear operator on ${\mathcal {D}}(U).$ Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.[42]

In the special case when $F$ is a diffeomorphism from an open subset $V$ of $\mathbb {R} ^{n}$ onto an open subset $U$ of $\mathbb {R} ^{n}$ change of variables under the integral gives

\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.

In this particular case, then, $F #$ is defined by the transpose formula:

\left\langle F^{\sharp }T,\phi \right\rangle =\left\langle T,\left|\det d(F^{-1})\right|\phi \circ F^{-1}\right\rangle .

Convolution

Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions. Recall that if $f$ and $g$ are functions on $\mathbb {R} ^{n}$ then we denote by $f\ast g$ the convolution of $f$ and $g$ , defined at $x\in \mathbb {R} ^{n}$ to be the integral

(f\ast g)(x):=\int _{\mathbb {R} ^{n}}f(x-y)g(y)\,dy=\int _{\mathbb {R} ^{n}}f(y)g(x-y)\,dy

provided that the integral exists. If $1\leq p,q,r\leq \infty$ are such that 1/r = (1/p) + (1/q) - 1 then for any functions $f\in L^{p}(\mathbb {R} ^{n})$ and $g\in L^{q}(\mathbb {R} ^{n})$ we have $f\ast g\in L^{r}(\mathbb {R} ^{n})$ and $\|f\ast g\|_{L^{r}}\leq \|f\|_{L^{p}}\|g\|_{L^{q}}.$ [43] If $f$ and $g$ are continuous functions on $\mathbb {R} ^{n},$ at least one of which has compact support, then $\operatorname {supp} (f\ast g)\subseteq \operatorname {supp} (f)+\operatorname {supp} (g)$ and if $A\subseteq \mathbb {R} ^{n}$ then the value of $f\ast g$ on $A$ do not depend on the values of $f$ outside of the Minkowski sum $A-\operatorname {supp} (g)=\{a-s:a\in A,s\in \operatorname {supp} (g)\}.$ [43]

Importantly, if $g\in L^{1}(\mathbb {R} ^{n})$ has compact support then for any $0\leq k\leq \infty ,$ the convolution map $f\mapsto f\ast g$ is continuous when considered as the map $C^{k}(\mathbb {R} ^{n})\to C^{k}(\mathbb {R} ^{n})$ or as the map $C_{c}^{k}(\mathbb {R} ^{n})\to C_{c}^{k}(\mathbb {R} ^{n}).$ [43]

Translation and symmetry

Given $a\in \mathbb {R} ^{n},$ the translation operator $τ a$ sends $f:\mathbb {R} ^{n}\to \mathbb {C}$ to $\tau _{a}f:\mathbb {R} ^{n}\to \mathbb {C} ,$ defined by $\tau _{a}f(y)=f(y-a).$ This can be extended by the transpose to distributions in the following way: given a distribution $T$ , the translation of $T$ by $a$ is the distribution $\tau _{a}T:{\mathcal {D}}(\mathbb {R} ^{n})\to \mathbb {C}$ defined by $\tau _{a}T(\phi ):=\left\langle T,\tau _{-a}\phi \right\rangle .$ [44][45]

Given $f:\mathbb {R} ^{n}\to \mathbb {C} ,$ define the function ${\tilde {f}}:\mathbb {R} ^{n}\to \mathbb {C}$ by ${\tilde {f}}(x):=f(-x).$ Given a distribution $T$ , let ${\tilde {T}}:{\mathcal {D}}(\mathbb {R} ^{n})\to \mathbb {C}$ be the distribution defined by ${\tilde {T}}(\phi ):=T\left({\tilde {\phi }}\right).$ The operator $T\mapsto {\tilde {T}}$ is called the symmetry with respect to the origin.[44]

Convolution of a test function with a distribution

Convolution with $f\in {\mathcal {D}}(\mathbb {R} ^{n})$ defines a linear map:

{\begin{cases}C_{f}:{\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n})\\C_{f}(g):=f\ast g\end{cases}}

which is continuous with respect to the canonical LF space topology on ${\mathcal {D}}(\mathbb {R} ^{n}).$

Convolution of $f$ with a distribution $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ can be defined by taking the transpose of C_f relative to the duality pairing of ${\mathcal {D}}(\mathbb {R} ^{n})$ with the space ${\mathcal {D}}'(\mathbb {R} ^{n})$ of distributions.[46] If $f,g,\phi \in {\mathcal {D}}(\mathbb {R} ^{n}),$ then by Fubini's theorem

\langle C_{f}g,\phi \rangle =\int _{\mathbb {R} ^{n}}\phi (x)\int _{\mathbb {R} ^{n}}f(x-y)g(y)\,dy\,dx=\left\langle g,C_{\tilde {f}}\phi \right\rangle .

Extending by continuity, the convolution of $f$ with a distribution $T$ is defined by

\langle f\ast T,\phi \rangle =\left\langle T,{\tilde {f}}\ast \phi \right\rangle ,

for all $\phi \in {\mathcal {D}}(\mathbb {R} ^{n}).$

An alternative way to define the convolution of a test function $f$ and a distribution $T$ is to use the translation operator $τ a$ . The convolution of the compactly supported function $f$ and the distribution $T$ is then the function defined for each $x\in \mathbb {R} ^{n}$ by

(f\ast T)(x)=\left\langle T,\tau _{x}{\tilde {f}}\right\rangle .

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution $T$ has compact support then if $f$ is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on $\mathbb {C} ^{n}$ to $\mathbb {R} ^{n},$ the restriction of an entire function of exponential type in $\mathbb {C} ^{n}$ to $\mathbb {R} ^{n}$ ) then the same is true of $T\ast f.$ [44] If the distribution $T$ has compact support as well, then $f\ast T$ is a compactly supported function, and the Titchmarsh convolution theorem Hörmander (1983, Theorem 4.3.3) implies that

\operatorname {ch} (\operatorname {supp} (f\ast T))=\operatorname {ch} (\operatorname {supp} (f))+\operatorname {ch} (\operatorname {supp} (T))

where ch denotes the convex hull and supp denotes the support.

Convolution of a smooth function with a distribution

Let $f\in C^{\infty }(\mathbb {R} ^{n})$ and $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ and assume that at least one of $f$ and $T$ has compact support. The convolution of $f$ and $T$ , denoted by $f\ast T$ or by $T\ast f,$ is the smooth function:[44]

{\begin{cases}f\ast T:\mathbb {R} ^{n}\to \mathbb {C} \\(f\ast T)(x):=\left\langle T,\tau _{x}{\tilde {f}}\right\rangle \end{cases}}

satisfying for all $p\in \mathbb {N} ^{n}$ :

{\begin{aligned}&\operatorname {supp} (f\ast T)\subseteq \operatorname {supp} (f)+\operatorname {supp} (T)\\[6pt]&{\text{ for all }}p\in \mathbb {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{aligned}}

If $T$ is a distribution then the map $f\mapsto T\ast f$ is continuous as a map ${\mathcal {D}}(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$ where if in addition $T$ has compact support then it is also continuous as the map $C^{\infty }(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$ and continuous as the map ${\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n}).$ [44]

If $L:{\mathcal {D}}(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$ is a continuous linear map such that $L\partial ^{\alpha }\phi =\partial ^{\alpha }L\phi$ for all $\alpha$ and all $\phi \in {\mathcal {D}}(\mathbb {R} ^{n})$ then there exists a distribution $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ such that $L\phi =T\circ \phi$ for all $\phi \in {\mathcal {D}}(\mathbb {R} ^{n}).$ [6]

Example.[6] Let H be the Heaviside function on $ℝ$ . For any $\phi \in {\mathcal {D}}(\mathbb {R} ),$

(H\ast \phi )(x)=\int _{-\infty }^{x}\phi (t)\,dt.

Let $\delta$ be the Dirac measure at 0 and $\delta '$ its derivative as a distribution. Then $\delta '\ast H=\delta$ and $1\ast \delta '=0.$ Importantly, the associative law fails to hold:

1=1\ast \delta =1\ast (\delta '\ast H)\neq (1\ast \delta ')\ast H=0\ast H=0.

Convolution of distributions

It is also possible to define the convolution of two distributions $S$ and $T$ on $\mathbb {R} ^{n},$ provided one of them has compact support. Informally, in order to define $S * T$ where $T$ has compact support, the idea is to extend the definition of the convolution $*$ to a linear operation on distributions so that the associativity formula

S\ast (T\ast \phi )=(S\ast T)\ast \phi

continues to hold for all test functions $\phi .$ [47]

It is also possible to provide a more explicit characterization of the convolution of distributions.[46] Suppose that $S$ and $T$ are distributions and that $S$ has compact support. Then the linear maps

{\begin{cases}\bullet \ast {\tilde {S}}:{\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n})\\f\mapsto f\ast {\tilde {S}}\end{cases}}\qquad {\begin{cases}\bullet \ast {\tilde {T}}:{\mathcal {D}}(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})\\f\mapsto f\ast {\tilde {T}}\end{cases}}

are continuous. The transposes of these maps,

{}^{t}\left(\bullet \ast {\tilde {S}}\right):{\mathcal {D}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})\qquad {}^{t}\left(\bullet \ast {\tilde {T}}\right):{\mathcal {E}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})

are consequently continuous and one may show that

{}^{t}\left(\bullet \ast {\tilde {S}}\right)(T)={}^{t}\left(\bullet \ast {\tilde {T}}\right)(S).

[44]

This common value is called the convolution of $S$ and $T$ and it is a distribution that is denoted by $S\ast T$ or $T\ast S.$ It satisfies $\operatorname {supp} (S\ast T)\subseteq \operatorname {supp} (S)+\operatorname {supp} (T).$ [44] If $S$ and $T$ are two distributions, at least one of which has compact support, then for any $a\in \mathbb {R} ^{n},$ $\tau _{a}(S\ast T)=\left(\tau _{a}S\right)\ast T=S\ast \left(\tau _{a}T\right).$ [44] If $T$ is a distribution in $\mathbb {R} ^{n}$ and if $\delta$ is a Dirac measure then $T\ast \delta =T.$ [44]

Suppose that it is $T$ that has compact support. For $\phi \in {\mathcal {D}}(\mathbb {R} ^{n})$ consider the function

\psi (x)=\langle T,\tau _{-x}\phi \rangle .

It can be readily shown that this defines a smooth function of $x$ , which moreover has compact support. The convolution of $S$ and $T$ is defined by

\langle S\ast T,\phi \rangle =\langle S,\psi \rangle .

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index $α$ ,

\partial ^{\alpha }(S\ast T)=(\partial ^{\alpha }S)\ast T=S\ast (\partial ^{\alpha }T).

The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.[44]

This definition of convolution remains valid under less restrictive assumptions about $S$ and $T$ .[48]

The convolution of distributions with compact support induces a continuous bilinear map ${\mathcal {E}}'\times {\mathcal {E}}'\to {\mathcal {E}}'$ defined by $(S,T)\mapsto S*T,$ where ${\mathcal {E}}'$ denotes the space of distributions with compact support.[14] However, the convolution map as a function ${\mathcal {E}}'\times {\mathcal {D}}'\to {\mathcal {D}}'$ is not continuous[14] although it is separately continuous.[49] The convolution maps ${\mathcal {D}}(\mathbb {R} ^{n})\times {\mathcal {D}}'\to {\mathcal {D}}'$ and ${\mathcal {D}}(\mathbb {R} ^{n})\times {\mathcal {D}}'\to {\mathcal {D}}(\mathbb {R} ^{n})$ given by $(f,T)\mapsto f*T$ both fail to be continuous.[14] Each of these non-continuous maps is, however, separately continuous and hypocontinuous.[14]

Convolution versus multiplication

In general, regularity is required for multiplication products and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let $F(\alpha )=f\in {\mathcal {O}}'_{C}$ be a rapidly decreasing tempered distribution or, equivalently, $F(f)=\alpha \in {\mathcal {O}}_{M}$ be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let $F$ be the normalized (unitary, ordinary frequency) Fourier transform [50] then, according to Schwartz (1951),

F(f*g)=F(f)\cdot F(g)

F(\alpha \cdot g)=F(\alpha )*F(g)

hold within the space of tempered distributions.[51][52][53] In particular, these equations become the Poisson Summation Formula if $g\equiv \operatorname {\text{Ш}}$ is the Dirac Comb.[54] The space of all rapidly decreasing tempered distributions is also called the space of convolution operators ${\mathcal {O}}'_{C}$ and the space of all ordinary functions within the space of tempered distributions is also called the space of multiplication operators ${\mathcal {O}}_{M}.$ More generally, $F({\mathcal {O}}'_{C})={\mathcal {O}}_{M}$ and $F({\mathcal {O}}_{M})={\mathcal {O}}'_{C}.$ [55][56] A particular case is the Paley-Wiener-Schwartz Theorem which states that $F({\mathcal {E}}')=\operatorname {PW}$ and $F(\operatorname {PW} )={\mathcal {E}}'.$ This is because ${\mathcal {E}}'\subseteq {\mathcal {O}}'_{C}$ and $\operatorname {PW} \subseteq {\mathcal {O}}_{M}.$ In other words, compactly supported tempered distributions ${\mathcal {E}}'$ belong to the space of convolution operators ${\mathcal {O}}'_{C}$ and Paley-Wiener functions $\operatorname {PW} ,$ better known as bandlimited functions, belong to the space of multiplication operators ${\mathcal {O}}_{M}.$ [57]

For example, let $g\equiv \operatorname {\text{Ш}} \in {\mathcal {S}}'$ be the Dirac comb and $f\equiv \delta \in {\mathcal {E}}'$ be the Dirac delta then $\alpha \equiv 1\in \operatorname {PW}$ is the function that is constantly one and both equations yield the Dirac comb identity. Another example is to let $g$ be the Dirac comb and $f\equiv \operatorname {rect} \in {\mathcal {E}}'$ be the rectangular function then $\alpha \equiv \operatorname {sinc} \in \operatorname {PW}$ is the sinc function and both equations yield the Classical Sampling Theorem for suitable $\operatorname {rect}$ functions. More generally, if $g$ is the Dirac comb and $f\in {\mathcal {S}}\subseteq {\mathcal {O}}'_{C}\cap {\mathcal {O}}_{M}$ is a smooth window function (Schwartz function), e.g. the Gaussian, then $\alpha \in {\mathcal {S}}$ is another smooth window function (Schwartz function). They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.

Tensor product of distributions

Let $U\subseteq \mathbb {R} ^{m}$ and $V\subseteq \mathbb {R} ^{n}$ be open sets. Assume all vector spaces to be over the field $\mathbb {F} ,$ where $\mathbb {F} =\mathbb {R}$ or $\mathbb {C} .$ For $f\in {\mathcal {D}}(U\times V)$ we define the following family of functions:

\left\{\left.{\begin{cases}f_{x}:V\to \mathbb {F} \\y\mapsto f(x,y)\end{cases}}\right|x\in U\right\},\qquad \left\{\left.{\begin{cases}f^{y}:U\to \mathbb {F} \\x\mapsto f(x,y)\end{cases}}\right|y\in V\right\}.

Given $S\in {\mathcal {D}}'(U)$ and $T\in {\mathcal {D}}'(V)$ we define the following functions:

{\begin{aligned}{\begin{cases}\langle S,f^{\bullet }\rangle :V\to \mathbb {F} \\y\mapsto \langle S,f^{y}\rangle \end{cases}}\\[8pt]{\begin{cases}\langle T,f_{\bullet }\rangle :U\to \mathbb {F} \\x\mapsto \langle T,f_{x}\rangle \end{cases}}\end{aligned}}

Note that $\langle T,f_{\bullet }\rangle \in {\mathcal {D}}(U)$ and $\langle S,f^{\bullet }\rangle \in {\mathcal {D}}(V).$ Now we define the following continuous linear maps associated to $S$ and $T$ :

{\begin{aligned}{\mathcal {D}}'(U)\ni S&\longrightarrow {\begin{cases}{\mathcal {D}}(U\times V)\to {\mathcal {D}}(V)\\f\mapsto \langle S,f^{\bullet }\rangle \end{cases}}\\[8pt]{\mathcal {D}}'(V)\ni T&\longrightarrow {\begin{cases}{\mathcal {D}}(U\times V)\to {\mathcal {D}}(U)\\f\mapsto \langle T,f_{\bullet }\rangle \end{cases}}\end{aligned}}

Moreover if either $S$ (resp. $T$ ) has compact support then it also induces a continuous linear map of $C^{\infty }(U\times V)\to C^{\infty }(V)$ (resp. $C^{\infty }(U\times V)\to C^{\infty }(U)$ ).[58]

Fubini's theorem for distributions[58] — Let $S\in {\mathcal {D}}'(U)$ and $T\in {\mathcal {D}}'(V).$ For every $f\in {\mathcal {D}}(U\times V)$ we have: $\langle S,\langle T,f_{\bullet }\rangle \rangle =\langle T,\langle S,f^{\bullet }\rangle \rangle .$

Definition. The tensor product of $S\in {\mathcal {D}}'(U)$ and $T\in {\mathcal {D}}'(V),$ denoted by $S\otimes T$ or $T\otimes S,$ is a distribution in $U\times V$ and is defined by:[58]

(S\otimes T)(f):=\langle S,\langle T,f_{\bullet }\rangle \rangle =\langle T,\langle S,f^{\bullet }\rangle \rangle .

Schwartz kernel theorem

The tensor product defines a bilinear map

{\begin{cases}{\mathcal {D}}'(U)\times {\mathcal {D}}'(V)\to {\mathcal {D}}'(U\times V)\\(S,T)\mapsto S\otimes T\end{cases}}

the span of the range of this map is a dense subspace of its codomain. Furthermore, $\operatorname {supp} (S\otimes T)=\operatorname {supp} (S)\times \operatorname {supp} (T).$ [58] Moreover $(S,T)\mapsto S\otimes T$ induces continuous bilinear maps:

{\begin{aligned}{\mathcal {E}}'(U)\times {\mathcal {E}}'(V)&\to {\mathcal {E}}'(U\times V)\\{\mathcal {S}}'(\mathbb {R} ^{m})\times {\mathcal {S}}'(\mathbb {R} ^{n})&\to {\mathcal {S}}'(\mathbb {R} ^{m+n})\end{aligned}}

where ${\mathcal {E}}'$ denotes the space of distributions with compact support and ${\mathcal {S}}$ is the Schwartz space of rapidly decreasing functions.[14]

Schwartz kernel theorem[59] — We have canonical TVS isomorphisms:

{\begin{aligned}{\mathcal {S}}'(\mathbb {R} ^{m+n})&\cong {\mathcal {S}}'(\mathbb {R} ^{m})\ {\widehat {\otimes }}\ {\mathcal {S}}'(\mathbb {R} ^{n})\cong L_{b}({\mathcal {S}}(\mathbb {R} ^{m});{\mathcal {S}}'(\mathbb {R} ^{n}))\\{\mathcal {E}}'(U\times V)&\cong {\mathcal {E}}'(U)\ {\widehat {\otimes }}\ {\mathcal {E}}'(V)\cong L_{b}(C^{\infty }(U);{\mathcal {E}}'(V))\\{\mathcal {D}}'(U\times V)&\cong {\mathcal {D}}'(U)\ {\widehat {\otimes }}\ {\mathcal {D}}'(V)\cong L_{b}({\mathcal {D}}(U);{\mathcal {D}}'(V))\end{aligned}}

Here ${\widehat {\otimes }}$ represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product, since these spaces are nuclear) and $L_{b}(X;Y)$ has the topology of uniform convergence on bounded subsets.

This result does not hold for Hilbert spaces such as $L^{2}$ and its dual space.[60] Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space $L^{2}$ ? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because ${\mathcal {D}}(U)$ is a nuclear space that the Schwartz kernel theorem holds.

Spaces of distributions

For all $0 < k < \infty$ and all $1 < p < \infty$ , all of the following canonical injections are continuous and have a range that is dense in their codomain:

{\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}}

where the topologies on $L_{c}^{q}(U)$ ( $1\leq q\leq \infty$ ) are defined as direct limits of the spaces $L_{c}^{q}(K)$ in a manner analogous to how the topologies on $C_{c}^{k}(U)$ were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in the codomain. Indeed, $C_{c}^{\infty }(U)$ is even sequentially dense in every $C_{c}^{k}(U).$ [28] All of the canonical injections $C_{c}^{\infty }(U)\to L^{p}(U)$ ( $1\leq p\leq \infty$ ) are continuous and the range of this injection is dense in the codomain if and only if $p\neq \infty$ (here $L^{p}(U)$ has its usual norm topology).[61]

Suppose that $X$ is one of the spaces $C_{c}^{k}(U)$ ( $k\in \{0,1,\ldots ,\infty \}$ ) or $L_{c}^{p}(U)$ ( $1\leq p\leq \infty$ ) or $L^{p}(U)$ ( $1\leq p<\infty$ ). Since the canonical injection $\operatorname {In} _{X}:C_{c}^{\infty }(U)\to X$ is a continuous injection whose image is dense in the codomain, the transpose ${}^{t}\operatorname {In} _{X}:X'_{b}\to {\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}$ is a continuous injection. This transpose thus allows us to identify $X'$ with a certain vector subspace of the space of distributions. This transpose map is not necessarily a TVS-embedding so that topology that this map transfers to the image $\operatorname {Im} \left({}^{t}\operatorname {In} _{X}\right)$ is finer than the subspace topology that this space inherits from ${\mathcal {D}}'(U).$ A linear subspace of ${\mathcal {D}}'(U)$ carrying a locally convex topology that is finer than the subspace topology induced by ${\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}$ is called a space of distributions.[61] Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order $\leq$ some integer, distributions induced by a positive Radon measure, distributions induced by an $L^{p}$ -function, etc.) and any representation theorem about the dual space of $X$ may, through the transpose ${}^{t}\operatorname {In} _{X}:X'_{b}\to {\mathcal {D}}'(U),$ be transferred directly to elements of the space $\operatorname {Im} \left({}^{t}\operatorname {In} _{X}\right).$

Radon measures

The natural inclusion $\operatorname {In} :C_{c}^{\infty }(U)\to C_{c}^{0}(U)$ is a continuous injection whose image is dense in its codomain, so the transpose ${}^{t}\operatorname {In$ is also a continuous injection.

Note that the continuous dual space $(C_{c}^{0}(U))'_{b}$ can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals $T\in (C_{c}^{0}(U))'_{b}$ and integral with respect to a Radon measure; that is,

if $T\in (C_{c}^{0}(U))'_{b}$ then there exists a Radon measure $\mu$ on $U$ such that for all $f\in C_{c}^{0}(U),T(f)=\textstyle \int _{U}f\,d\mu ,$ and

if $\mu$ is a Radon measure on $U$ then the linear functional on $C_{c}^{0}(U)$ defined by $C_{c}^{0}(U)\ni f\mapsto \textstyle \int _{U}f\,d\mu$ is continuous.

Through the injection ${}^{t}\operatorname {In$ every Radon measure becomes a distribution on $U$ . If $f$ is a locally integrable function on $U$ then the distribution $\phi \mapsto \textstyle \int _{U}f(x)\phi (x)\,dx$ is a Radon measure; so Radon measures form a large and important space of distributions.

The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally $L^{\infty }$ functions in $U$ :

Theorem.[34] Suppose

T\in {\mathcal {D}}'(U)

is a Radon measure,

V \subseteq U

is a neighborhood of the support of

T

, and

I=\{p\in \mathbb {N} ^{n}:|p|\leq k\}.

There exists is a family of locally

L^{\infty }

functions in

U

such that

T=\sum _{|p|\leq k}\partial ^{p}\mu _{p}

and for very

p\in I,\operatorname {supp} \mu _{p}\subseteq V.

Positive Radon measures

A linear function $T$ on a space of functions is called positive if whenever a function $f$ that belongs to the domain of $T$ is non-negative (i.e. $f$ is real-valued and $f\geq 0$ ) then $T(f)\geq 0.$ One may show that every positive linear functional on $C_{c}^{0}(U)$ is necessarily continuous (i.e. necessarily a Radon measure).[62]Note that Lebesgue measure is an example of a positive Radon measure.

Locally integrable functions as distributions

One particularly important class of Radon measures are those that are induced locally integrable functions. The function $f:U\to \mathbb {R}$ is called locally integrable if it is Lebesgue integrable over every compact subset $K$ of $U$ .[note 18] This is a large class of functions which includes all continuous functions and all L^p functions. The topology on ${\mathcal {D}}(U)$ is defined in such a fashion that any locally integrable function $f$ yields a continuous linear functional on ${\mathcal {D}}(U)$ – that is, an element of ${\mathcal {D}}'(U)$ – denoted here by $T f$ , whose value on the test function $\phi$ is given by the Lebesgue integral:

\langle T_{f},\phi \rangle =\int _{U}f\phi \,dx.

Conventionally, one abuses notation by identifying $T f$ with $f,$ provided no confusion can arise, and thus the pairing between $T f$ and $\phi$ is often written

\langle f,\phi \rangle =\langle T_{f},\phi \rangle .

If $f$ and $g$ are two locally integrable functions, then the associated distributions $T f$ and $T g$ are equal to the same element of ${\mathcal {D}}'(U)$ if and only if $f$ and $g$ are equal almost everywhere (see, for instance, Hörmander (1983, Theorem 1.2.5)). In a similar manner, every Radon measure $\mu$ on $U$ defines an element of ${\mathcal {D}}'(U)$ whose value on the test function $\phi$ is $\textstyle \int \phi \,d\mu .$ As above, it is conventional to abuse notation and write the pairing between a Radon measure $\mu$ and a test function $\phi$ as $\langle \mu ,\phi \rangle .$ Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

Test functions as distributions

The test functions are themselves locally integrable, and so define distributions. The space of test functions $C_{c}^{\infty }(U)$ is sequentially dense in ${\mathcal {D}}'(U)$ with respect to the strong topology on ${\mathcal {D}}'(U).$ [29] This means that for any $T\in {\mathcal {D}}'(U),$ there is a sequence of test functions, $(\phi _{i})_{i=1}^{\infty },$ that converges to $T\in {\mathcal {D}}'(U)$ (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,

\langle \phi _{i},\psi \rangle \to \langle T,\psi \rangle \qquad {\text{ for all }}\psi \in {\mathcal {D}}(U).

Furthermore, $C_{c}^{\infty }(U)$ is also sequentially dense in the strong dual space of $C^{\infty }(U).$ [29]

Distributions with compact support

The natural inclusion $\operatorname {In} :C_{c}^{\infty }(U)\to C^{\infty }(U)$ is a continuous injection whose image is dense in its codomain, so the transpose ${}^{t}\operatorname {In$ is also a continuous injection. Thus the image of the transpose, denoted by ${\mathcal {E}}'(U),$ forms a space of distributions when it is endowed with the strong dual topology of $(C^{\infty }(U))'_{b}$ (transferred to it via the transpose map ${}^{t}\operatorname {In$ so the topology of ${\mathcal {E}}'(U)$ is finer than the subspace topology that this set inherits from ${\mathcal {D}}'(U)$ ).[31]

The elements of ${\mathcal {E}}'(U)=(C^{\infty }(U))'_{b}$ can be identified as the space of distributions with compact support.[31] Explicitly, if $T$ is a distribution on $U$ then the following are equivalent,

$T\in {\mathcal {E}}'(U)$ ;

the support of $T$ is compact;

the restriction of $T$ to $C_{c}^{\infty }(U),$ when that space is equipped with the subspace topology inherited from $C^{\infty }(U)$ (a coarser topology than the canonical LF topology), is continuous;[31]

there is a compact subset $K$ of $U$ such that for every test function $\phi$ whose support is completely outside of $K$ , we have $T(\phi )=0.$

Compactly supported distributions define continuous linear functionals on the space $C^{\infty }(U)$ ; recall that the topology on $C^{\infty }(U)$ is defined such that a sequence of test functions $\phi _{k}$ converges to 0 if and only if all derivatives of $\phi _{k}$ converge uniformly to 0 on every compact subset of $U$ . 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 $C_{c}^{\infty }(U)$ to $C^{\infty }(U).$

Distributions of finite order

Let $k\in \mathbb {N} .$ The natural inclusion $\operatorname {In} :C_{c}^{\infty }(U)\to C_{c}^{k}(U)$ is a continuous injection whose image is dense in its codomain, so the transpose ${}^{t}\operatorname {In$ is also a continuous injection. Consequently, the image of ${}^{t}\operatorname {In} ,$ denoted by ${\mathcal {D}}'^{k}(U),$ forms a space of distributions when it is endowed with the strong dual topology of $(C_{c}^{k}(U))'_{b}$ (transferred to it via the transpose map ${}^{t}\operatorname {In$ so ${\mathcal {D}}'^{m}(U)$ 's topology is finer than the subspace topology that this set inherits from ${\mathcal {D}}'(U)$ ). The elements of ${\mathcal {D}}'^{k}(U)$ are the distributions of order $\leq k$ .[34] The distributions of order ≤ 0, which are also called distributions of order $0$ , are exactly the distributions that are Radon measures (described above).

For $0\neq k\in \mathbb {N} ,$ a distribution of order $k$ is a distribution of order $\leq k$ that is not a distribution of order $\leq k - 1$ .[34]

A distribution is said to be of finite order if there is some integer $k$ such that it is a distribution of order $\leq k$ , and the set of distributions of finite order is denoted by ${\mathcal {D}}'^{F}(U).$ Note that if $k \leq l$ then ${\mathcal {D}}'^{k}(U)\subseteq {\mathcal {D}}'^{l}(U)$ so that ${\mathcal {D}}'^{F}(U)$ is a vector subspace of ${\mathcal {D}}'(U)$ and furthermore, if and only if ${\mathcal {D}}'^{F}(U)={\mathcal {D}}'(U).$ [34]

Structure of distributions of finite order

Every distribution with compact support in $U$ is a distribution of finite order.[34] Indeed, every distribution in $U$ is locally a distribution of finite order, in the following sense:[34] If $V$ is an open and relatively compact subset of $U$ and if $\rho _{VU}$ is the restriction mapping from $U$ to $V$ , then the image of ${\mathcal {D}}'(U)$ under $\rho _{VU}$ is contained in ${\mathcal {D}}'^{F}(V).$

The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:

Theorem.[34] Suppose

T\in {\mathcal {D}}'(U)

has finite order

\leq k

and

I=\{p\in \mathbb {N} ^{n}:|p|\leq k\}.

Given any open subset

V

of

U

containing the support of

T

, there is a family of Radon measures in

U

,

(\mu _{p})_{p\in I},

such that for very

p\in I,\operatorname {supp} (\mu _{p})\subseteq V

and

T=\sum _{|p|\leq k}\partial ^{p}\mu _{p}.

Example. (Distributions of infinite order) Let $U := (0, \infty)$ and for every test function $f,$ let

Sf:=\sum _{m=1}^{\infty }(\partial ^{m}f)\left({\frac {1}{m}}\right).

Then $S$ is a distribution of infinite order on $U$ . Moreover, $S$ can not be extended to a distribution on $ℝ$ ; that is, there exists no distribution $T$ on $ℝ$ such that the restriction of $T$ to $U$ is equal to $T$ .[63]

Tempered distributions and Fourier transform

Defined below are the tempered distributions, which form a subspace of ${\mathcal {D}}'(\mathbb {R} ^{n}),$ the space of distributions on $\mathbb {R} ^{n}.$ This is a proper subspace: while every tempered distribution is a distribution and an element of ${\mathcal {D}}'(\mathbb {R} ^{n}),$ the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distributions in ${\mathcal {D}}'(\mathbb {R} ^{n}).$

Schwartz space

The Schwartz space, ${\mathcal {S}}(\mathbb {R} ^{n}),$ is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus ${\displaystyle \phi$ is in the Schwartz space provided that any derivative of $\phi ,$ multiplied with any power of |x|, converges to 0 as $| x | \to \infty$ . These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices $\alpha$ and $\beta$ define:

p_{\alpha ,\beta }(\phi )~=~\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }\partial ^{\beta }\phi (x)\right|.

Then $\phi$ is in the Schwartz space if all the values satisfy:

p_{\alpha ,\beta }(\phi )<\infty .

The family of seminorms $p α, β$ defines a locally convex topology on the Schwartz space. For n = 1, the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:[64]

|f|_{m,k}=\sup _{|p|\leq m}\left(\sup _{x\in \mathbb {R} ^{n}}\left\{(1+|x|)^{k}\left|(\partial ^{\alpha }f)(x)\right|\right\}\right),\qquad k,m\in \mathbb {N} .

Otherwise, one can define a norm on ${\mathcal {S}}(\mathbb {R} ^{n})$ via

\|\phi \|_{k}~=~\max _{|\alpha |+|\beta |\leq k}\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }\partial ^{\beta }\phi (x)\right|,\qquad k\geq 1.

The Schwartz space is a Fréchet space (i.e. a complete metrizable locally convex space). Because the Fourier transform changes $\partial ^{\alpha }$ into multiplication by $x^{\alpha }$ and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

A sequence $\{f_{i}\}$ in ${\mathcal {S}}(\mathbb {R} ^{n})$ converges to 0 in ${\mathcal {S}}(\mathbb {R} ^{n})$ if and only if the functions $(1+|x|)^{k}(\partial ^{p}f_{i})(x)$ converge to 0 uniformly in the whole of $\mathbb {R} ^{n},$ which implies that such a sequence must converge to zero in $C^{\infty }(\mathbb {R} ^{n}).$ [64]

${\mathcal {D}}(\mathbb {R} ^{n})$ is dense in ${\mathcal {S}}(\mathbb {R} ^{n}).$ The subset of all analytic Schwartz functions is dense in ${\mathcal {S}}(\mathbb {R} ^{n})$ as well.[65]

The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms

{\mathcal {S}}(\mathbb {R} ^{m})\ {\widehat {\otimes }}\ {\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{m+n}),

where ${\widehat {\otimes }}$ represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product).[59]

Tempered distributions

The natural inclusion $\operatorname {In$ is a continuous injection whose image is dense in its codomain, so the transpose ${}^{t}\operatorname {In$ is also a continuous injection. Thus, the image of the transpose map, denoted by ${\mathcal {S}}'(\mathbb {R} ^{n}),$ forms a space of distributions when it is endowed with the strong dual topology of $({\mathcal {S}}(\mathbb {R} ^{n}))'_{b}$ (transferred to it via the transpose map ${}^{t}\operatorname {In$ so the topology of ${\mathcal {S}}'(\mathbb {R} ^{n})$ is finer than the subspace topology that this set inherits from ${\mathcal {D}}'(\mathbb {R} ^{n})$ ).

The space ${\mathcal {S}}'(\mathbb {R} ^{n})$ is called the space of >tempered distributions is it is the continuous dual of the Schwartz space. Equivalently, a distribution $T$ is a tempered distribution if and only if

\left({\text{ for all }}\alpha ,\beta \in \mathbb {N} ^{n}:\lim _{m\to \infty }p_{\alpha ,\beta }(\phi _{m})=0\right)\Longrightarrow \lim _{m\to \infty }T(\phi _{m})=0.

The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of $L^{p}(\mathbb {R} ^{n})$ for $p \geq 1$ are tempered distributions.

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

Fourier transform

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform $F:{\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})$ is a TVS-automorphism of the Schwartz space, and the Fourier transform is defined to be its transpose ${}^{t}F:{\mathcal {S}}'(\mathbb {R} ^{n})\to {\mathcal {S}}'(\mathbb {R} ^{n}),$ which (abusing notation) will again be denoted by $F$ . So the Fourier transform of the tempered distribution $T$ is defined by $(FT)(ψ) = T (Fψ)$ for every Schwartz function $ψ$ . $FT$ is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that

F{\dfrac {dT}{dx}}=ixFT

and also with convolution: if $T$ is a tempered distribution and $ψ$ is a slowly increasing smooth function on $\mathbb {R} ^{n},$ $ψT$ is again a tempered distribution and

F(\psi T)=F\psi *FT

is the convolution of $FT$ and $Fψ$ . In particular, the Fourier transform of the constant function equal to 1 is the $δ$ distribution.

Expressing tempered distributions as sums of derivatives

If $T\in {\mathcal {S}}'(\mathbb {R} ^{n})$ is a tempered distribution, then there exists a constant $C > 0$ , and positive integers $M$ and $N$ such that for all Schwartz functions $\phi \in {\mathcal {S}}(\mathbb {R} ^{n})$

\langle T,\phi \rangle \leq C\sum \nolimits _{|\alpha |\leq N,|\beta |\leq M}\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }\partial ^{\beta }\phi (x)\right|=C\sum \nolimits _{|\alpha |\leq N,|\beta |\leq M}p_{\alpha ,\beta }(\phi ).

This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function $F$ and a multi-index $α$ such that

T=\partial ^{\alpha }F.

Restriction of distributions to compact sets

If $T\in {\mathcal {D}}'(\mathbb {R} ^{n}),$ then for any compact set $K\subseteq \mathbb {R} ^{n},$ there exists a continuous function $F$ compactly supported in $\mathbb {R} ^{n}$ (possibly on a larger set than $K$ itself) and a multi-index $α$ such that $T=\partial ^{\alpha }F$ on $C_{c}^{\infty }(K).$

Using holomorphic functions as test functions

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

Notes

$D_{f}$ turns out to also be linear and continuous when the space of test functions is given a certain topology called the canonical LF topology.
The Schwartz space consists of smooth rapidly decreasing test functions, where "rapidly decreasing" means that the function decreases faster than any polynomial increases as points in its domain move away from the origin.
Except for the trivial (i.e. identically $0$ ) map, which of course is always analytic.
Note that $i$ being an integer implies $i \neq \infty$ . This is sometimes expressed as $0 \leq i < k + 1$ . Since $\infty + 1 = \infty$ , the inequality " $0 \leq i < k + 1$ " means: $0 \leq i < \infty$ if $k = \infty$ , while if $k \neq \infty$ then it means $0 \leq i \leq k$ .
The image of the compact set $K$ under a continuous $ℝ$ -valued map (e.g. $x\mapsto \left|\partial ^{p}f(x)\right|$ for $x \in U$ ) is itself a compact, and thus bounded, subset of $ℝ$ . If $K \neq \emptyset$ then this implies that each of the functions defined above is $ℝ$ -valued (i.e. none of the supremums above are ever equal to $\infty$ ).
If we take $𝕂$ to be the set of all compact subsets of $U$ then we can use the universal property of direct limits to conclude that the inclusion $\operatorname {In} _{K}^{U}:C^{k}(K)\to C_{c}^{k}(U)$ is a continuous and even that they are topological embedding for every compact subset $K \subseteq U$ . If however, we take $𝕂$ to be the set of closures of some countable increasing sequence of relatively compact open subsets of $U$ having all of the properties mentioned earlier in this in this article then we immediately deduce that $C_{c}^{k}(U)$ is a Hausdorff locally convex strict LF-space (and even a strict LB-space when $k \neq \infty$ ). All of these facts can also be proved directly without using direct systems (although with more work).
For any TVS $X$ (metrizable or otherwise), the notion of completeness depends entirely on a certain so-called "canonical uniformity" that is defined using only the subtraction operation (see the article Complete topological vector space for more details). In this way, the notion of a complete TVS does not require the existence of any metric. However, if the TVS $X$ is metrizable and if $d$ is any translation-invariant metric on $X$ that defines its topology, then $X$ is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if the $(X, d)$ is a complete metric space. So if a TVS $X$ happens to have a topology that can be defined by such a metric $d$ then $d$ may be used to deduce the completeness of $X$ but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the Cartesian product of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS $\mathbb {R} ^{\mathbb {N} },$ which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on $\mathbb {R} ^{\mathbb {N} }$ ).
One reason for giving $C_{c}^{\infty }(U)$ the canonical LF topology is because it is with this topology that $C_{c}^{\infty }(U)$ and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of topological vector spaces (TVSs) that are particularly similar to finite-dimensional Euclidean spaces: the Banach spaces (especially Hilbert spaces) and the nuclear Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are nuclear Montel Hilbert spaces but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. infinite differentiability, such as $C_{c}^{\infty }(U)$ and $C^{\infty }(U)$ ) end up being nuclear TVSs while TVSs associated with finite continuous differentiability (such as $C^{k}(K)$ with $K$ compact and $k \neq \infty$ ) often end up being non-nuclear spaces, such as Banach spaces.
Even though the topology of $C_{c}^{\infty }(U)$ is not metrizable, a linear functional on $C_{c}^{\infty }(U)$ is continuous if and only if it is sequentially continuous.
A null sequence is a sequence that converges to the origin.
A sequence $x • = (x i) \infty i =1$ is said to be Mackey convergent to $0$ in $X,$ if there exists a divergent sequence $r • = (r i) \infty i =1 \to \infty$ of positive real number such that $(r i x i) \infty i =1$ is a bounded set in $X.$
If ${\mathcal {P}}$ is also directed under the usual function comparison then we can take the finite collection to consist of a single element.
In functional analysis, the strong dual topology is often the "standard" or "default" topology placed on the continuous dual space $X',$ where if $X$ is a normed space then this strong dual topology is the same as the usual norm-induced topology on $X'.$
Technically, the topology must be coarser than the strong dual topology and also simultaneously be finer that the weak* topology.
Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences.
This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.
For example let $U=\mathbb {R}$ and take $P$ to be the ordinary derivative for functions of one real variable and assume the support of $\phi$ to be contained in the finite interval $(a,b),$ then since $\operatorname {supp} (\phi )\subseteq (a,b)$
${\begin{aligned}\int _{\mathbb {R} }\phi '(x)f(x)\,dx&=\int _{a}^{b}\phi '(x)f(x)\,dx\\&=\phi (x)f(x){\big \vert }_{a}^{b}-\int _{a}^{b}f'(x)\phi (x)\,dx\\&=\phi (b)f(b)-\phi (a)f(a)-\int _{a}^{b}f'(x)\phi (x)\,dx\\&=\int _{a}^{b}f'(x)\phi (x)\,dx\end{aligned}}$
where the last equality is because $\phi (a)=\phi (b)=0.$
For more information on such class of functions, see the entry on locally integrable functions.

References

Trèves 2006, pp. 222-223.
See for example Grubb 2009, p. 14.
Trèves 2006, pp. 85-89.
Trèves 2006, pp. 142-149.
Trèves 2006, pp. 356-358.
Rudin 1991, pp. 149-181.
Trèves 2006, pp. 131-134.
Trèves 2006, pp. 247-252.
Trèves 2006, pp. 126-134.
Trèves 2006, pp. 136-148.
Trèves 2006, pp. 245-247.
Rudin 1991, pp. 149-155.
Narici & Beckenstein 2011, pp. 446-447.
Trèves 2006, p. 423.
See for example Grubb 2009, p. 14.
See for example Schaefer & Wolff 1999, p. 173.
Gabriyelyan, Saak "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces" (2017)
Gabriyelyan, S.S. Kakol J., and·Leiderman, A. "The strong Pitkeev property for topological groups and topological vector spaces"
Trèves 2006, pp. 195-201.
Narici & Beckenstein 2011, p. 435.
Trèves 2006, pp. 526-534.
Trèves 2006, p. 357.
"Topological vector space". Encyclopedia of Mathematics. Encyclopedia of Mathematics. Retrieved September 6, 2020. It is a Montel space, hence paracompact, and so normal.
T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.
According to Gel'fand & Shilov 1966–1968, v. 1, §1.2
Trèves 2006, pp. 351-359.
Narici & Beckenstein 2011, pp. 441-457.
Trèves 2006, pp. 150-160.
Trèves 2006, pp. 300-304.
Trèves 2006, pp. 253-255.
Trèves 2006, pp. 255-257.
Trèves 2006, pp. 264-266.
Rudin 1991, p. 165.
Trèves 2006, pp. 258-264.
Rudin 1991, pp. 169-170.
Strichartz 1994, §2.3; Trèves 2006.
Rudin 1991, p. 180.
Trèves 2006, p. 261.
Lyons, T. (1998). "Differential equations driven by rough signals". Revista Matemática Iberoamericana: 215–310. doi:10.4171/RMI/240.
Hairer, Martin (2014). "A theory of regularity structures". Inventiones Mathematicae. 198 (2): 269–504. arXiv:1303.5113. Bibcode:2014InMat.198..269H. doi:10.1007/s00222-014-0505-4.
See for example Hörmander 1983, Theorem 6.1.1.
See Hörmander 1983, Theorem 6.1.2.
Trèves 2006, pp. 278-283.
Trèves 2006, pp. 284-297.
See for example Rudin 1991, §6.29.
Trèves 2006, Chapter 27.
Hörmander 1983, §IV.2 proves the uniqueness of such an extension.
See for instance Gel'fand & Shilov 1966–1968, v. 1, pp. 103–104 and Benedetto 1997, Definition 2.5.8.
Trèves 2006, p. 294.
Folland, G.B. (1989). Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press.
Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.
Woodward, P.M. (1953). Probability and Information Theory with Applications to Radar. Oxford, UK: Pergamon Press.
Trèves 2006, pp. 318-319.
Friedlander, F.G.; Joshi, M.S. (1998). Introduction to the Theory of Distributions. Cambridge, UK: Cambridge University Press.
Schwartz 1951.
Trèves 2006, pp. 416-419.
Trèves 2006, p. 531.
Trèves 2006, pp. 509-510.
Trèves 2006, pp. 240-252.
Trèves 2006, p. 218.
Rudin 1991, pp. 177-181.
Trèves 2006, pp. 92-94.
Trèves 2006, pp. 160.

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Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

Distribution (mathematics)

History

Notation

Definitions of test functions and distributions

Topology on Ck(U)

Topology on Ck(K)

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

Topology on the spaces of test functions and distributions

Canonical LF topology

Basic properties

Distributions

Topology on the space of distributions

Topological properties

Convergent sequences

Localization of distributions

Restrictions to an open subset

Gluing and distributions that vanish in a set

Support of a distribution

Distributions with compact support

Global structure of distributions

Decomposition of distributions as sums of derivatives of continuous functions

Operations on distributions

Preliminaries: Transpose of a linear operator

Differentiation of distributions

Differential operators acting on smooth functions

Multiplication of distributions by smooth functions

Problem of multiplying distributions

Composition with a smooth function

Convolution

Convolution of a test function with a distribution

Convolution of a smooth function with a distribution

Convolution of distributions

Convolution versus multiplication

Tensor product of distributions

Schwartz kernel theorem

Spaces of distributions

Radon measures

Locally integrable functions as distributions

Distributions with compact support

Distributions of finite order

Tempered distributions and Fourier transform

Using holomorphic functions as test functions

See also

Notes

References

Bibliography

Further reading

Topology on C^k(U)

Topology on C^k(K)

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