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

Notation:

Let $k\in \{0,1,2,\ldots ,\infty \}.$
Let $C^{k}(U)$ denote the vector space of all $k$ -times continuously differentiable real or complex-valued functions on $U$ .
For any compact subset $K\subseteq U,$ $K\subseteq U,$ let $C^{k}(K)$ $C^{k}(K)$ and $C^{k}(K;U)$ $C^{k}(K;U)$ both denote the vector space of all those functions $f\in C^{k}(U)$ $f\in C^{k}(U)$ such that $\operatorname {supp} (f)\subseteq K.$ $\operatorname {supp} (f)\subseteq K.$
- If $f\in C^{k}(K)$ then the domain of $f$ is $U$ and not $K$ . So although $C^{k}(K)$ depends on both $K$ and $U$ , only $K$ is typically indicated. The justification for this common practice is detailed below. The notation $C^{k}(K;U)$ will only be used when the notation $C^{k}(K)$ risks being ambiguous.
- Every $C^{k}(K)$ contains the constant $0$ map, even if $K=\varnothing .$
Let $C_{c}^{k}(U)$ $C_{c}^{k}(U)$ denote the set of all $f\in C^{k}(U)$ $f\in C^{k}(U)$ such that $f\in C^{k}(K)$ $f\in C^{k}(K)$ 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 $K\subseteq U$ ranges over all compact subsets of $U.$
- 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 also a complex-valued test function on $U.$

Thumb — The graph of the bump function $(x,y)\in \mathbb {R} ^{2}\mapsto \Psi (r),$ where $r=\left(x^{2}+y^{2}\right)^{\frac {1}{2}}$ 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.

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 functions on

U

. We will use both

{\mathcal {D}}(U)

and

C_{c}^{\infty }(U)

to denote this space.

Distributions on $U$ are continuous linear functionals on $C_{c}^{\infty }(U)$ when this vector space is endowed with a particular topology called the canonical LF-topology. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on $C_{c}^{\infty }(U)$ that are often straightforward to verify.

Proposition: A linear functional $T$ on $C_{c}^{\infty }(U)$ is continuous, and therefore a distribution, if and only if any of the following equivalent conditions is satisfied:

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

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

is an integer such that

0\leq i\leq k

^{[note 1]} and

p

is a multi-index with length

|p|\leq k.

For

K\neq \varnothing

and

f\in C^{k}(U),

define:

${\begin{alignedat}{4}{\text{ (1) }}\ &s_{p,K}(f)&&:=\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\\[4pt]{\text{ (2) }}\ &q_{i,K}(f)&&:=\sup _{|p|\leq i}\left(\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\right)=\sup _{|p|\leq i}\left(s_{p,K}(f)\right)\\[4pt]{\text{ (3) }}\ &r_{i,K}(f)&&:=\sup _{\stackrel {|p|\leq i}{x_{0}\in K}}\left|\partial ^{p}f(x_{0})\right|\\[4pt]{\text{ (4) }}\ &t_{i,K}(f)&&:=\sup _{x_{0}\in K}\left(\sum _{|p|\leq i}\left|\partial ^{p}f(x_{0})\right|\right)\end{alignedat}}$

while for

K=\varnothing ,

define all the functions above to be the constant

0

map.

All of the functions above are non-negative $\mathbb {R}$ -valued^{[note 2]} seminorms on $C^{k}(U).$ As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.

Each of the following sets of seminorms ${\begin{alignedat}{4}A~:=\quad &\{q_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\B~:=\quad &\{r_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\C~:=\quad &\{t_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\D~:=\quad &\{s_{p,K}&&:\;K{\text{ compact and }}\;&&p\in \mathbb {N} ^{n}{\text{ satisfies }}\;&&|p|\leq k\}\end{alignedat}}$ generate the same locally convex vector topology on $C^{k}(U)$ (so for example, the topology generated by the seminorms in $A$ is equal to the topology generated by those in $C$ ).

The vector space

C^{k}(U)

is endowed with the locally convex topology induced by any one of the four families

A,B,C,D

of seminorms described above. This topology is also equal to the vector topology induced by all of the seminorms in

A\cup B\cup C\cup D.

With this topology, $C^{k}(U)$ becomes a locally convex Fréchet space that is not normable. Every element of $A\cup B\cup C\cup D$ is a continuous seminorm 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 compact $K,$ the net of partial derivatives $\left(\partial ^{p}f_{i}\right)_{i\in I}$ converges uniformly to $\partial ^{p}f$ on $K.$ ^[3] For any $k\in \{0,1,2,\ldots ,\infty \},$ any (von Neumann) 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]

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 on it by $C^{i}(U).$

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).

If $k$ is finite then $C^{k}(K)$ is a Banach space^[6] 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.^[6]

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

Suppose $U$ is an open subset of $\mathbb {R} ^{n}$ and $K\subseteq U$ is a compact subset. By definition, elements of $C^{k}(K)$ are functions with domain $U$ (in symbols, $C^{k}(K)\subseteq C^{k}(U)$ ), so the space $C^{k}(K)$ and its topology depend on $U;$ to make this dependence on the open set $U$ clear, temporarily denote $C^{k}(K)$ by $C^{k}(K;U).$ Importantly, changing the set $U$ to a different open subset $U'$ (with $K\subseteq U'$ ) will change the set $C^{k}(K)$ from $C^{k}(K;U)$ to $C^{k}(K;U'),$ ^{[note 3]} so that elements of $C^{k}(K)$ will be functions with domain $U'$ instead of $U.$ Despite $C^{k}(K)$ depending on the open set ( $U{\text{ or }}U'$ ), the standard notation for $C^{k}(K)$ makes no mention of it. This is justified because, as this subsection will now explain, the space $C^{k}(K;U)$ is canonically identified as a subspace of $C^{k}(K;U')$ (both algebraically and topologically).

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

Given $f\in C_{c}^{k}(U),$ its trivial extension to $V$ is the function $F:V\to \mathbb {C}$ defined by: $F(x)={\begin{cases}f(x)&x\in U,\\0&{\text{otherwise}}.\end{cases}}$ This trivial extension belongs to $C^{k}(V)$ (because $f\in C_{c}^{k}(U)$ has compact support) and it will be denoted by $I(f)$ (that is, $I(f):=F$ ). The assignment $f\mapsto I(f)$ thus induces a map $I:C_{c}^{k}(U)\to C^{k}(V)$ 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 $K\subseteq U$ (where $K$ is also a compact subset of $V$ since $K\subseteq U\subseteq V$ ), ${\begin{alignedat}{4}I\left(C^{k}(K;U)\right)&~=~C^{k}(K;V)\qquad {\text{ and thus }}\\I\left(C_{c}^{k}(U)\right)&~\subseteq ~C_{c}^{k}(V).\end{alignedat}}$ If $I$ is restricted to $C^{k}(K;U)$ then the following induced linear map is a homeomorphism (linear homeomorphisms are called TVS-isomorphisms): ${\begin{alignedat}{4}\,&C^{k}(K;U)&&\to \,&&C^{k}(K;V)\\&f&&\mapsto \,&&I(f)\\\end{alignedat}}$ and thus the next map is a topological embedding: ${\begin{alignedat}{4}\,&C^{k}(K;U)&&\to \,&&C^{k}(V)\\&f&&\mapsto \,&&I(f).\\\end{alignedat}}$ Using the injection $I:C_{c}^{k}(U)\to C^{k}(V)$ the vector space $C_{c}^{k}(U)$ is canonically identified 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 $C^{k}(V).$ Thus the topology on $C^{k}(K;U)$ is independent of the open subset $U$ of $\mathbb {R} ^{n}$ that contains $K,$ ^[7] which justifies the practice of writing $C^{k}(K)$ instead of $C^{k}(K;U).$

Canonical LF topology

Recall that $C_{c}^{k}(U)$ denotes all 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 all compact subsets of $U.$ Moreover, for each $k,\,C_{c}^{k}(U)$ 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).

Unless indicated otherwise, it is endowed with a topology called the canonical LF topology, whose definition is given in the article: Spaces of test functions and distributions.

The canonical LF-topology is not metrizable and importantly, it is strictly finer than the subspace topology that $C^{\infty }(U)$ induces on $C_{c}^{\infty }(U).$ However, the canonical LF-topology does make $C_{c}^{\infty }(U)$ into a complete reflexive nuclear^[8] Montel^[9] bornological barrelled Mackey space; the same is true of its strong dual space (that is, the space of all distributions with its usual topology). The canonical LF-topology can be defined in various ways.

Distributions

As discussed earlier, continuous linear functionals on a $C_{c}^{\infty }(U)$ are known as distributions on $U.$ Other equivalent definitions are described below.

By definition, a distribution on $U$ is 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.

There is a 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;
$T$ 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 $f\in C_{c}^{\infty }(U),$ ${\textstyle \lim _{i\to \infty }T\left(f_{i}\right)=T(f);}$ ^{[note 4]}
T is sequentially continuous at the origin; in other words, T maps null sequences^{[note 5]} 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), ${\textstyle \lim _{i\to \infty }T\left(f_{i}\right)=0;}$
- a null sequence is by definition any 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 convergent null sequences 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 $f_{\bullet }=\left(f_{i}\right)_{i=1}^{\infty }$ is said to be Mackey convergent to the origin if there exists a divergent sequence $r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }\to \infty$ of positive real numbers such that the sequence $\left(r_{i}f_{i}\right)_{i=1}^{\infty }$ is bounded; every sequence that is Mackey convergent to the origin 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 closed;
There exists a continuous seminorm $g$ on $C_{c}^{\infty }(U)$ such that $|T|\leq g;$
There exists a constant $C>0$ and a finite subset $\{g_{1},\ldots ,g_{m}\}\subseteq {\mathcal {P}}$ (where ${\mathcal {P}}$ is any collection of continuous seminorms that defines the canonical LF topology on $C_{c}^{\infty }(U)$ ) such that $|T|\leq C(g_{1}+\cdots +g_{m});$ ^{[note 6]}
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 $K,$ ^[10] $|T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N_{K}\};$
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 $T(f_{i})\to 0;$

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

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 ${\mathcal {D}}'(U).$ Importantly, unless indicated otherwise, the topology on ${\mathcal {D}}'(U)$ is the strong dual topology; if the topology is instead the weak-* topology then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes ${\mathcal {D}}'(U)$ into a complete nuclear space, to name just a few of its desirable properties.

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

A linear map from ${\mathcal {D}}'(U)$ into another locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general topological spaces (for example, that are not also locally convex topological vector spaces). The same is true of maps from $C_{c}^{\infty }(U)$ (more generally, this is true of maps from any locally convex bornological space).