Spaces of test functions and distributions

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (July 2024) (Learn how and when to remove this message)

In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset ${\displaystyle U\subseteq \mathbb {R} ^{n}}$ that have compact support. The space of all test functions, denoted by ${\displaystyle C_{c}^{\infty }(U),}$ is endowed with a certain topology, called the canonical LF-topology, that makes ${\displaystyle C_{c}^{\infty }(U)}$ into a complete Hausdorff locally convex TVS. The strong dual space of ${\displaystyle C_{c}^{\infty }(U)}$ is called the space of distributions on ${\displaystyle U}$ and is denoted by ${\displaystyle {\mathcal {D}}^{\prime }(U):=\left(C_{c}^{\infty }(U)\right)_{b}^{\prime },}$ where the "${\displaystyle b}$" subscript indicates that the continuous dual space of ${\displaystyle C_{c}^{\infty }(U),}$ denoted by ${\displaystyle \left(C_{c}^{\infty }(U)\right)^{\prime },}$ is endowed with the strong dual topology.

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 1]} as test functions gives rise to a certain subspace of ${\displaystyle {\mathcal {D}}^{\prime }(U)}$ 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 ${\displaystyle {\mathcal {D}}^{\prime }(U)}$ 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 2]} Use of analytic test functions leads to Sato's theory of hyperfunctions.

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).

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.

Note that for all ${\displaystyle j,k\in \{0,1,2,\ldots ,\infty \}}$ and any compact subsets K and L of U, we have: $${\displaystyle {\begin{aligned}C^{k}(K)&\subseteq C_{c}^{k}(U)\subseteq C^{k}(U)\\C^{k}(K)&\subseteq C^{k}(L)&&{\text{ if }}K\subseteq L\\C^{k}(K)&\subseteq C^{j}(K)&&{\text{ if }}j\leq k\\C_{c}^{k}(U)&\subseteq C_{c}^{j}(U)&&{\text{ if }}j\leq k\\C^{k}(U)&\subseteq C^{j}(U)&&{\text{ if }}j\leq k\\\end{aligned}}}$$

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

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 equivalent conditions are satisfied:

1. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C>0}$ and ${\displaystyle N\in \mathbb {N} }$ (dependent on ${\displaystyle K}$) 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 ${\displaystyle C>0}$ and ${\displaystyle N\in \mathbb {N} }$ such that for all ${\displaystyle f\in C_{c}^{\infty }(U)}$ with support contained in ${\displaystyle K,}$^[2] $${\displaystyle |T(f)|\leq C\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N\}.}$$
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 locally convex topological vector spaces (TVSs) be defined first. First, a (non-normable) topology on ${\displaystyle C^{\infty }(U)}$ will be defined, then every ${\displaystyle C^{\infty }(K)}$ will be endowed with the subspace topology induced on it by ${\displaystyle C^{\infty }(U),}$ and finally the (non-metrizable) canonical LF-topology on ${\displaystyle C_{c}^{\infty }(U)}$ will be defined. The space of distributions, being defined as the continuous dual space of ${\displaystyle C_{c}^{\infty }(U),}$ is then endowed with the (non-metrizable) strong dual topology induced by ${\displaystyle C_{c}^{\infty }(U)}$ and the canonical LF-topology (this topology is a generalization of the usual operator norm induced topology that is placed on the continuous dual spaces of normed spaces). This finally permits consideration of more advanced notions such as convergence of distributions (both sequences and nets), various (sub)spaces of distributions, and operations on distributions, including extending differential equations to distributions.

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

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

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

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.

Suppose ${\displaystyle k\in \{0,1,2,\ldots ,\infty \}}$ and ${\displaystyle K}$ is an arbitrary compact subset of ${\displaystyle U.}$ Suppose ${\displaystyle i}$ an integer such that ${\displaystyle 0\leq i\leq k}$^{[note 3]} and ${\displaystyle p}$ is a multi-index with length ${\displaystyle |p|\leq k.}$ For ${\displaystyle K\neq \varnothing ,}$ define:

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

while for ${\displaystyle K=\varnothing ,}$ define all the functions above to be the constant 0 map.

All of the functions above are non-negative ${\displaystyle \mathbb {R} }$-valued^{[note 4]} seminorms on ${\displaystyle C^{k}(U).}$ 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 $${\displaystyle {\begin{alignedat}{4}A~:=\quad &\{q_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\B~:=\quad &\{r_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\C~:=\quad &\{t_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\D~:=\quad &\{s_{p,K}&&:\;K{\text{ compact and }}\;&&p\in \mathbb {N} ^{n}{\text{ satisfies }}\;&&|p|\leq k\}\end{alignedat}}}$$ generate the same locally convex vector topology on ${\displaystyle C^{k}(U)}$ (so for example, the topology generated by the seminorms in ${\displaystyle A}$ is equal to the topology generated by those in ${\displaystyle C}$).

The vector space ${\displaystyle C^{k}(U)}$ is endowed with the locally convex topology induced by any one of the four families ${\displaystyle A,B,C,D}$ of seminorms described above. This topology is also equal to the vector topology induced by all of the seminorms in ${\displaystyle A\cup B\cup C\cup D.}$

With this topology, ${\displaystyle C^{k}(U)}$ becomes a locally convex Fréchet space that is not normable. Every element of ${\displaystyle A\cup B\cup C\cup D}$ is a continuous seminorm 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 ${\displaystyle p}$ with ${\displaystyle |p|<k+1}$ and every compact ${\displaystyle K,}$ the net of partial derivatives ${\displaystyle \left(\partial ^{p}f_{i}\right)_{i\in I}}$ converges uniformly to ${\displaystyle \partial ^{p}f}$ on ${\displaystyle K.}$^[3] For any ${\displaystyle k\in \{0,1,2,\ldots ,\infty \},}$ any (von Neumann) 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 ${\displaystyle k=\infty .}$^[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 ${\displaystyle W}$ of ${\displaystyle C^{\infty }(U)}$ is open in this topology if and only if there exists ${\displaystyle i\in \mathbb {N} }$ such that ${\displaystyle W}$ is open when ${\displaystyle C^{\infty }(U)}$ is endowed with the subspace topology induced on it by ${\displaystyle C^{i}(U).}$

If the family of compact sets ${\displaystyle \mathbb {K} =\left\{{\overline {U}}_{1},{\overline {U}}_{2},\ldots \right\}}$ satisfies ${\textstyle U=\bigcup _{j=1}^{\infty }U_{j}}$ and ${\displaystyle {\overline {U}}_{i}\subseteq U_{i+1}}$ for all ${\displaystyle i,}$ 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 defining families of seminorms (A through D). For example, using the seminorms ${\displaystyle (r_{i,K_{i}})_{i=1}^{\infty }}$ results in the metric $${\displaystyle d(f,g):=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {r_{i,{\overline {U}}_{i}}(f-g)}{1+r_{i,{\overline {U}}_{i}}(f-g)}}=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {\sup _{|p|\leq i,x\in {\overline {U}}_{i}}\left|\partial ^{p}(f-g)(x)\right|}{\left[1+\sup _{|p|\leq i,x\in {\overline {U}}_{i}}\left|\partial ^{p}(f-g)(x)\right|\right]}}.}$$

Often, it is easier to just consider seminorms (avoiding any metric) and use the tools of functional analysis.

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

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

For any compact subset ${\displaystyle K\subseteq U,}$ ${\displaystyle C^{k}(K)}$ is a closed subspace of the Fréchet space ${\displaystyle C^{k}(U)}$ and is thus also a Fréchet space. For all compact ${\displaystyle K,L\subseteq U}$ satisfying ${\displaystyle K\subseteq L,}$ denote the inclusion map by ${\displaystyle \operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L).}$ Then this map is a linear embedding of TVSs (that is, it is a linear map that is also a topological embedding) whose image (or "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 ${\displaystyle k=2,}$ 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).

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 ${\displaystyle V}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$ containing ${\displaystyle U}$ and for any compact subset ${\displaystyle K\subseteq V,}$ let ${\displaystyle C^{k}(K;V)}$ is the vector subspace of ${\displaystyle C^{k}(V)}$ consisting of maps with support contained in ${\displaystyle K.}$ Given ${\displaystyle f\in C_{c}^{k}(U),}$ its trivial extension to V is by definition, the function ${\displaystyle I(f):=F:V\to \mathbb {C} }$ 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 ${\displaystyle K\subseteq U}$ (where ${\displaystyle K}$ is also a compact subset of ${\displaystyle V}$ since ${\displaystyle K\subseteq U\subseteq V}$) we have $${\displaystyle {\begin{alignedat}{4}I\left(C^{k}(K;U)\right)&~=~C^{k}(K;V)\qquad {\text{ and thus }}\\I\left(C_{c}^{k}(U)\right)&~\subseteq ~C_{c}^{k}(V)\end{alignedat}}}$$ If I is restricted to ${\displaystyle C^{k}(K;U)}$ then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism): $${\displaystyle {\begin{alignedat}{4}\,&C^{k}(K;U)&&\to \,&&C^{k}(K;V)\\&f&&\mapsto \,&&I(f)\\\end{alignedat}}}$$ and thus the next two maps (which like the previous map are defined by ${\displaystyle f\mapsto I(f)}$) are topological embeddings: $${\displaystyle C^{k}(K;U)\to C^{k}(V),\qquad {\text{ and }}\qquad C^{k}(K;U)\to C_{c}^{k}(V),}$$ (the topology on ${\displaystyle C_{c}^{k}(V)}$ is the canonical LF topology, which is defined later). Using the injection $${\displaystyle I:C_{c}^{k}(U)\to C^{k}(V)}$$ the vector space ${\displaystyle C_{c}^{k}(U)}$ is canonically identified with its image in ${\displaystyle C_{c}^{k}(V)\subseteq C^{k}(V)}$ (however, if ${\displaystyle U\neq V}$ then ${\displaystyle I:C_{c}^{\infty }(U)\to C_{c}^{\infty }(V)}$ is not a topological embedding when these spaces are endowed with their canonical LF topologies, although it is continuous).^[8] 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 writing ${\displaystyle C^{k}(K)}$ instead of ${\displaystyle C^{k}(K;U).}$

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

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

This section defines 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 ${\displaystyle K\leq L}$ if and only if ${\displaystyle K\subseteq L,}$ which in particular makes the collection ${\displaystyle \mathbb {K} }$ of compact subsets of U into a directed set (we say that such a collection is directed by subset inclusion). For all compact ${\displaystyle K,L\subseteq U}$ satisfying ${\displaystyle K\subseteq L,}$ there are inclusion maps $${\displaystyle \operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L)\quad {\text{and}}\quad \operatorname {In} _{K}^{U}:C^{k}(K)\to C_{c}^{k}(U).}$$

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 ${\displaystyle \mathbb {K} }$ (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 ${\displaystyle \operatorname {In} _{\bullet }^{U}=(\operatorname {In} _{K}^{U})_{K\in \mathbb {K} }}$ continuous.

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

As is common in mathematics literature, the space ${\displaystyle C_{c}^{k}(U)}$ is henceforth assumed to be endowed with its canonical LF topology (unless explicitly stated otherwise).

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:

For all ${\displaystyle K\in \mathbb {K} ,}$ ${\displaystyle U\cap C^{k}(K)}$ is a neighborhood of the origin in ${\displaystyle C^{k}(K).}$

CN

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.

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 ${\displaystyle P.}$^[9]

For any ${\displaystyle 1\leq k\leq \infty ,}$ the canonical LF topology on ${\displaystyle C_{c}^{k}(U)}$ is the weakest locally convex TVS topology making all linear differential operators in ${\displaystyle U}$ of order ${\displaystyle \,<k+1}$ into continuous maps from ${\displaystyle C_{c}^{k}(U)}$ into ${\displaystyle C_{c}^{0}(U).}$^[9]

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 ${\displaystyle \mathbb {K} }$ of compact sets. And by considering different collections ${\displaystyle \mathbb {K} }$ (in particular, those ${\displaystyle \mathbb {K} }$ 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 ${\displaystyle k\neq \infty }$), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).^{[note 5]}

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 ${\displaystyle K\in \mathbb {K} ,}$ the restriction of u to ${\displaystyle C^{k}(K)}$ is continuous (or bounded).^[10][11]

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.^[8] If (and only if) ${\displaystyle U\neq V}$ then ${\displaystyle I\left(C_{c}^{\infty }(U)\right)}$ 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.^[8] Consequently, if ${\displaystyle U\neq V}$ then the transpose of ${\displaystyle I:C_{c}^{\infty }(U)\to C_{c}^{\infty }(V)}$ is neither one-to-one nor onto.^[8]

A subset ${\displaystyle B\subseteq C_{c}^{k}(U)}$ is bounded in ${\displaystyle C_{c}^{k}(U)}$ if and only if there exists some ${\displaystyle K\in \mathbb {K} }$ such that ${\displaystyle B\subseteq C^{k}(K)}$ and ${\displaystyle B}$ is a bounded subset of ${\displaystyle C^{k}(K).}$^[11] Moreover, if ${\displaystyle K\subseteq U}$ is compact and ${\displaystyle S\subseteq C^{k}(K)}$ then ${\displaystyle S}$ is bounded in ${\displaystyle C^{k}(K)}$ if and only if it is bounded in ${\displaystyle C^{k}(U).}$ For any ${\displaystyle 0\leq k\leq \infty ,}$ 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 ${\displaystyle \infty +1=\infty .}$^[11]

For all compact ${\displaystyle K\subseteq U,}$ 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 6]} for an explanation of how the non-metrizable space ${\displaystyle C_{c}^{k}(U)}$ can be complete even though 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 7]}

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 explained 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]).

• The differentiation map ${\displaystyle C_{c}^{\infty }(U)\to C_{c}^{\infty }(U)}$ is a 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]

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 ${\displaystyle {\mathcal {D}}^{\prime }(U).}$

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