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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 ${\displaystyle f}$ is normally thought of as acting on the points in its domain by "sending" a point x in its domain to the point ${\displaystyle f(x).}$ Instead of acting on points, distribution theory reinterprets functions such as ${\displaystyle f}$ 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 ${\displaystyle f:\mathbb {R} \to \mathbb {R} ,}$ 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 ${\displaystyle f}$ "acts on" a test function g by "sending" g to the number ${\displaystyle \textstyle \int _{\mathbb {R} }fg\,dx.}$ This new action of ${\displaystyle f}$ is thus a complex (or real)-valued map, denoted by ${\displaystyle D_{f},}$ 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 ${\displaystyle \mathbb {R} .}$ 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 ${\displaystyle U\subset \mathbb {R} ^{n}.}$ This space of test functions is denoted by ${\displaystyle C_{c}^{\infty }(U)}$ or ${\displaystyle {\mathcal {D}}(U)}$ and a distribution on U is by definition a linear functional on ${\displaystyle C_{c}^{\infty }(U)}$ that is continuous when ${\displaystyle C_{c}^{\infty }(U)}$ is given a topology called the canonical LF topology. This leads to the space of (all) distributions on U, usually denoted by ${\displaystyle {\mathcal {D}}'(U)}$ (note the prime), which by definition is the space of all distributions on ${\displaystyle U}$ (that is, it is the continuous dual space of ${\displaystyle C_{c}^{\infty }(U)}$); 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 ${\displaystyle U=\mathbb {R} ^{n}}$ then the use of Schwartz functions^{[note 2]} as test functions gives rise to a certain subspace of ${\displaystyle {\mathcal {D}}'(U)}$ 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 ${\displaystyle {\mathcal {D}}'(U)}$ 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 ${\displaystyle C_{c}^{\infty }(U),}$ 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:

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

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

• ${\displaystyle k}$ will denote a non-negative integer or ${\displaystyle \infty .}$

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

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

${\displaystyle \langle f,g\rangle :=\int _{U}f(x)g(x)\,dx.}$

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

${\displaystyle {\begin{aligned}x^{\alpha }&=x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}\\\partial ^{\alpha }&={\frac {\partial ^{|\alpha |}}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}\end{aligned}}}$

We also introduce a partial order of all multi-indices by ${\displaystyle \beta \geq \alpha }$ if and only if ${\displaystyle \beta _{i}\geq \alpha _{i}}$ for all ${\displaystyle 1\leq i\leq n.}$ When ${\displaystyle \beta \geq \alpha }$ we define their multi-index binomial coefficient as:

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

• ${\displaystyle \mathbb {K} }$ will denote a certain non-empty collection of compact subsets of ${\displaystyle U}$ (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 ${\displaystyle \mathbb {R} ^{n}}$ with any (paracompact) smooth manifold.

The graph of the bump function ${\displaystyle (x,y)\in \mathbf {R} ^{2}\mapsto \Psi (r),}$ where ${\displaystyle r=(x^{2}+y^{2})^{\frac {1}{2}}}$ and ${\displaystyle \Psi (r)=e^{-{\frac {1}{1-r^{2}}}}\cdot \mathbf {1} _{\{|r|<1\}}.}$ This function is a test function on ${\displaystyle \mathbb {R} ^{2}}$ and is an element of ${\displaystyle C_{c}^{\infty }\left(\mathbb {R} ^{2}\right).}$ The support of this function is the closed unit disk in ${\displaystyle \mathbb {R} ^{2}.}$ It is non-zero on the open unit disk and it is equal to 0 everywhere outside of it.

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

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

Distributions on U are defined to be the continuous linear functionals on ${\displaystyle C_{c}^{\infty }(U)}$ 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 ${\displaystyle C_{c}^{\infty }(U)}$ then the T is a distribution if and only if the following are equivalent conditions are satisfied:

1. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C>0}$ and ${\displaystyle N\in \mathbb {N} }$ such that for all ${\displaystyle f\in C^{\infty }(K),}$^[1]

   ${\displaystyle |T(f)|\leq C\sup\{|\partial ^{\alpha }f(x)|:x\in U,|\alpha |\leq N\};}$

2. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C_{K}>0}$ and ${\displaystyle N_{K}\in \mathbb {N} }$ such that for all ${\displaystyle f\in C_{c}^{\infty }(U)}$ with support contained in ${\displaystyle K,}$^[2]

   ${\displaystyle |T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N_{K}\};}$

3. For any compact subset ${\displaystyle K\subseteq U}$ and any sequence ${\displaystyle \{f_{i}\}_{i=1}^{\infty }}$ in ${\displaystyle C^{\infty }(K),}$ if ${\displaystyle \{\partial ^{\alpha }f_{i}\}_{i=1}^{\infty }}$ converges uniformly to zero on ${\displaystyle K}$ for all multi-indices ${\displaystyle \alpha }$, then ${\displaystyle T(f_{i})\to 0.}$

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 ${\displaystyle C_{c}^{\infty }(U)}$ and ${\displaystyle {\mathcal {D}}(U).}$ 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 ${\displaystyle C^{\infty }(U),}$ then assign every ${\displaystyle C^{\infty }(K)}$ the subspace topology induced on it by ${\displaystyle C^{\infty }(U),}$ and finally we define the canonical LF-topology on ${\displaystyle C_{c}^{\infty }(U).}$ 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) ${\displaystyle U=\cup _{K\in \mathbb {K} }K,}$ and (2) for any compact K ? U there exists some K₂ ? K such that K ? K₂. The most common choices for K are:

• The set of all compact subsets of U, or
• A set ${\displaystyle \left\{{\overline {U_{1}}},{\overline {U_{2}}},\ldots \right\}}$ where ${\displaystyle U=\cup _{i=1}^{\infty }U_{i},}$ and for all i, ${\displaystyle {\overline {U_{i}}}\subseteq U_{i+1}}$ 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 ${\displaystyle \mathbb {R} ^{n},}$ is compact).

We make K into a directed set by defining K₁K₂ if and only if K₁ ? K₂. 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 K₁ and K₂ are any two such collections of compact subsets of U, then the topologies defined on ${\displaystyle C^{k}(U)}$ and ${\displaystyle C_{c}^{k}(U)}$ by using K₁ in place of K are the same as those defined by using K₂ in place of K.

Topology on C^k(U)

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

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

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

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

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

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

Metric defining the topology

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

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

Often, it is easier to just consider seminorms.

Topology on C^k(K)

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

For any compact subset K ? U, ${\displaystyle C^{k}(K)}$ is a closed subspace of the Fréchet space ${\displaystyle C^{k}(U)}$ and is thus also a Fréchet space. For all compact K, L ? U with K ? L, denote the natural inclusion by ${\displaystyle \operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L).}$ 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 ${\displaystyle C^{k}(K)}$ is identical to the subspace topology it inherits from ${\displaystyle C^{k}(L),}$ and also ${\displaystyle C^{k}(K)}$ is a closed subset of ${\displaystyle C^{k}(L).}$ The interior of ${\displaystyle C^{\infty }(K)}$ relative to ${\displaystyle C^{\infty }(U)}$ is empty.^[6]

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

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

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

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

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

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

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

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

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

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

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

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

Topology on the spaces of test functions and distributions

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

Canonical LF topology

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

For any two sets K and L, we declare that K L if and only if K ? 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 ? U with K ? L, there are natural inclusions

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

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

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

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 ${\displaystyle (C_{c}^{k}(U),\operatorname {In} _{\bullet }^{U})}$ where ${\displaystyle \operatorname {In} _{\bullet }^{U}:=\left(\operatorname {In} _{K}^{U}\right)_{K\in \mathbb {K} }}$ are the natural inclusions and where ${\displaystyle C_{c}^{k}(U)}$ is now endowed with the (unique) strongest locally convex topology making all of the inclusion maps ${\displaystyle \operatorname {In} _{\bullet }^{U}=\left(\operatorname {In} _{K}^{U}\right)_{K\in \mathbb {K} }}$ continuous.

Neighborhoods of the origin

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

Note that any convex set satisfying this condition is necessarily absorbing in ${\displaystyle C_{c}^{k}(U).}$ 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

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

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

For any 1 k , the canonical LF topology on ${\displaystyle C_{c}^{k}(U)}$ is the weakest locally convex TVS topology making all linear differential operators in U of order < k + 1 into continuous maps from ${\displaystyle C_{c}^{k}(U)}$ into ${\displaystyle C_{c}^{0}(U).}$^[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 ${\displaystyle C_{c}^{k}(U)}$ into a Hausdorff locally convex strict LF-space (and also a strict LB-space if k ? ?), 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 ${\displaystyle u:C_{c}^{k}(U)\to Y}$ 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 ? K, the restriction of u to ${\displaystyle C^{k}(K)}$ is continuous (or bounded).^[9][10]

Dependence of the canonical LF topology on U

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

Bounded subsets

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

Non-metrizability

For all compact K ? U, the interior of ${\displaystyle C^{k}(K)}$ in ${\displaystyle C_{c}^{k}(U)}$ is empty so that ${\displaystyle C_{c}^{k}(U)}$ is of the first category in itself. It follows from Baire's theorem that ${\displaystyle C_{c}^{k}(U)}$ is not metrizable and thus also not normable (see this footnote^{[note 7]} for an explanation of how the non-metrizable space ${\displaystyle C_{c}^{k}(U)}$ can be complete even thought it does not admit a metric). The fact that ${\displaystyle C_{c}^{\infty }(U)}$ is a nuclear Montel space makes up for the non-metrizability of ${\displaystyle C_{c}^{\infty }(U)}$ (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 ${\displaystyle \operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L)}$ are all topological embedding, one may show that all of the maps ${\displaystyle \operatorname {In} _{K}^{U}:C^{k}(K)\to C_{c}^{k}(U)}$ are also topological embeddings. Said differently, the topology on ${\displaystyle C^{k}(K)}$ is identical to the subspace topology that it inherits from ${\displaystyle C_{c}^{k}(U),}$ where recall that ${\displaystyle C^{k}(K)}$'s topology was defined to be the subspace topology induced on it by ${\displaystyle C^{k}(U).}$ In particular, both ${\displaystyle C_{c}^{k}(U)}$ and ${\displaystyle C^{k}(U)}$ induces the same subspace topology on ${\displaystyle C^{k}(K).}$ However, this does not imply that the canonical LF topology on ${\displaystyle C_{c}^{k}(U)}$ is equal to the subspace topology induced on ${\displaystyle C_{c}^{k}(U)}$ by ${\displaystyle C^{k}(U)}$; these two topologies on ${\displaystyle C_{c}^{k}(U)}$ are in fact never equal to each other since the canonical LF topology is never metrizable while the subspace topology induced on it by ${\displaystyle C^{k}(U)}$ is metrizable (since recall that ${\displaystyle C^{k}(U)}$ is metrizable). The canonical LF topology on ${\displaystyle C_{c}^{k}(U)}$ is actually strictly finer than the subspace topology that it inherits from ${\displaystyle C^{k}(U)}$ (thus the natural inclusion ${\displaystyle C_{c}^{k}(U)\to C^{k}(U)}$ is continuous but not a topological embedding).^[7]

Indeed, the canonical LF topology is so fine that if ${\displaystyle C_{c}^{\infty }(U)\to X}$ denotes some linear map that is a "natural inclusion" (such as ${\displaystyle C_{c}^{\infty }(U)\to C^{k}(U),}$ or ${\displaystyle C_{c}^{\infty }(U)\to L^{p}(U),}$ 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 ${\displaystyle C_{c}^{\infty }(U),}$ the fine nature of the canonical LF topology means that more linear functionals on ${\displaystyle C_{c}^{\infty }(U)}$ end up being continuous ("more" means as compared to a coarser topology that we could have placed on ${\displaystyle C_{c}^{\infty }(U)}$ such as for instance, the subspace topology induced by some ${\displaystyle C^{k}(U),}$ which although it would have made ${\displaystyle C_{c}^{\infty }(U)}$ metrizable, it would have also resulted in fewer linear functionals on ${\displaystyle C_{c}^{\infty }(U)}$ being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making ${\displaystyle C_{c}^{\infty }(U)}$ into a complete TVS^[12]).

Other properties

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

Distributions

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

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

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

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

Characterizations of distributions

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

1. T is a distribution;
2. (definition) T is continuous;
3. T is continuous at the origin;
4. T is uniformly continuous;
5. T is a bounded operator;
6. T is sequentially continuous;
   • explicitly, for every sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U)}$ that converges in ${\displaystyle C_{c}^{\infty }(U)}$ to some ${\displaystyle f\in C_{c}^{\infty }(U),}$ ${\displaystyle \lim _{i\to \infty }T\left(f_{i}\right)=T(f);}$^{[note 9]}
7. T is sequentially continuous at the origin; in other words, T maps null sequences^{[note 10]} to null sequences;
   • explicitly, for every sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U)}$ that converges in ${\displaystyle C_{c}^{\infty }(U)}$ to the origin (such a sequence is called a null sequence), ${\displaystyle \lim _{i\to \infty }T\left(f_{i}\right)=0;}$
   • a null sequence is by definition a sequence that converges to the origin;
8. T maps null sequences to bounded subsets;
   • explicitly, for every sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U)}$ that converges in ${\displaystyle C_{c}^{\infty }(U)}$ to the origin, the sequence ${\displaystyle \left(T\left(f_{i}\right)\right)_{i=1}^{\infty }}$ is bounded;
9. T maps Mackey convergence null sequences^{[note 11]} to bounded subsets;
   • explicitly, for every Mackey convergent null sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U),}$ the sequence ${\displaystyle \left(T\left(f_{i}\right)\right)_{i=1}^{\infty }}$ is bounded;
   • a sequence f_o = (f_i)^?
     _i=1 is said to be Mackey convergent to 0 if there exists a divergent sequence r_o = (r_i)^?
     _i=1 -> ? of positive real number such that the sequence (r_if_i)^?
     _i=1 is bounded; every sequence that is Mackey convergent to 0 necessarily converges to the origin (in the usual sense);
10. The kernel of T is a closed subspace of ${\displaystyle C_{c}^{\infty }(U);}$
11. The graph of T is a closed;
12. There exists a continuous seminorm g on ${\displaystyle C_{c}^{\infty }(U)}$ such that ${\displaystyle |T|\leq g;}$
13. There exists a constant C > 0, a collection of continuous seminorms, ${\displaystyle {\mathcal {P}},}$ that defines the canonical LF topology of ${\displaystyle C_{c}^{\infty }(U),}$ and a finite subset ${\displaystyle \{g_{1},\ldots ,g_{m}\}\subseteq {\mathcal {P}}}$ such that ${\displaystyle |T|\leq C(g_{1}+\cdots g_{m});}$^{[note 12]}
14. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C>0}$ and ${\displaystyle N\in \mathbb {N} }$ such that for all ${\displaystyle f\in C^{\infty }(K),}$^[1]

    ${\displaystyle |T(f)|\leq C\sup\{|\partial ^{p}f(x)|:x\in U,|\alpha |\leq N\};}$

15. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C_{K}>0}$ and ${\displaystyle N_{K}\in \mathbb {N} }$ such that for all ${\displaystyle f\in C_{c}^{\infty }(U)}$ with support contained in ${\displaystyle K,}$^[15]

    ${\displaystyle |T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N_{K}\};}$

16. For any compact subset ${\displaystyle K\subseteq U}$ and any sequence ${\displaystyle \{f_{i}\}_{i=1}^{\infty }}$ in ${\displaystyle C^{\infty }(K),}$ if ${\displaystyle \{\partial ^{p}f_{i}\}_{i=1}^{\infty }}$ converges uniformly to zero for all multi-indices p, then ${\displaystyle T(f_{i})\to 0;}$
17. 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

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

The three spaces ${\displaystyle C_{c}^{\infty }(U),}$ ${\displaystyle C^{\infty }(U),}$ and the Schwartz space ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n}),}$ 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 ${\displaystyle C^{\infty }(U)}$ and ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$ are both distinguished Fréchet spaces. Moreover, both ${\displaystyle C_{c}^{\infty }(U)}$ and ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$ are Schwartz TVSs.

Convergent sequences

Convergent sequences and their insufficiency to describe topologies

The strong dual spaces of ${\displaystyle C^{\infty }(U)}$ and ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$ are sequential spaces but not Fréchet-Urysohn spaces.^[17] Moreover, neither the space of test functions ${\displaystyle C_{c}^{\infty }(U)}$ nor its strong dual ${\displaystyle {\mathcal {D}}'(U)}$ 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 ${\displaystyle (f_{i})_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{k}(U)}$ converges in ${\displaystyle C_{c}^{k}(U)}$ if and only if there exists some K ? K such that ${\displaystyle C^{k}(K)}$ contains this sequence and this sequence converges in ${\displaystyle C^{k}(K)}$; equivalently, it converges if and only if the following two conditions hold:^[25]

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

Neither the space ${\displaystyle C_{c}^{\infty }(U)}$ nor its strong dual ${\displaystyle {\mathcal {D}}'(U)}$ 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 ${\displaystyle C_{c}^{\infty }(U).}$ The same can be said of the strong dual topology on ${\displaystyle {\mathcal {D}}'(U).}$