spaces of test functions and distributions

The following notation will be used throughout this article:

• $n$ is a fixed positive integer and $U$ is a fixed non-empty open subset of Euclidean space $\backslash R^\{n\}.$
• $\backslash N\; =\; \backslash \{0,\; 1,\; 2,\; \backslash ldots\backslash \}$ denotes the natural numbers.
• $k$ will denote a non-negative integer or $\backslash infty.$
• If $f$ is a function then $\backslash operatorname\{Dom\}(f)$ will denote its domain and the {{em|Support (mathematics)}} of $f,$ denoted by $\backslash operatorname\{supp\}(f),$ is defined to be the closure of the set $\backslash \{x\; \backslash in\; \backslash operatorname\{Dom\}(f):\; f(x)\; \backslash neq\; 0\backslash \}$ in $\backslash operatorname\{Dom\}(f).$
• For two functions $f,\; g\; :\; U\; \backslash to\; \backslash Complex$, the following notation defines a canonical pairing: $$\backslash langle\; f,\; g\backslash rangle\; :=\; \backslash int\_U\; f(x)\; g(x)\; \backslash ,dx.$$
• A {{em|multi-index}} of size $n$ is an element in $\backslash N^n$ (given that $n$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be $n$). The {{em|length}} of a multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$ is defined as $\backslash alpha\_1+\backslash cdots+\backslash alpha\_n$ and denoted by $|\backslash alpha|.$ Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$: $$\backslash begin\{align\}$$

x^\alpha &= x_1^{\alpha_1} \cdots x_n^{\alpha_n} \\

\partial^\alpha &= \frac{\partial^

\end{align} We also introduce a partial order of all multi-indices by $\backslash beta\; \backslash geq\; \backslash alpha$ if and only if $\backslash beta\_i\; \backslash geq\; \backslash alpha\_i$ for all $1\; \backslash leq\; i\backslash leq\; n.$ When $\backslash beta\; \backslash geq\; \backslash alpha$ we define their multi-index binomial coefficient as: $$\backslash binom\{\backslash beta\}\{\backslash alpha\}\; :=\; \backslash binom\{\backslash beta\_1\}\{\backslash alpha\_1\}\; \backslash cdots\; \backslash binom\{\backslash beta\_n\}\{\backslash alpha\_n\}.$$

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

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

{{block indent|em=1.5|text=Notation:

1. Let $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \}.$
2. Let $C^k(U)$ denote the vector space of all {{mvar|k}}-times continuously differentiable real or complex-valued functions on {{mvar|U}}.
3. For any compact subset $K\; \backslash subseteq\; U,$ let $C^k(K)$ and $C^k(K;U)$ both denote the vector space of all those functions $f\; \backslash in\; C^k(U)$ such that $\backslash operatorname\{supp\}(f)\; \backslash subseteq\; K.$
4. * If $f\; \backslash in\; C^k(K)$ then the domain of $f$ is {{mvar|U}} and not {{mvar|K}}. So although $C^k(K)$ depends on both {{mvar|K}} and {{mvar|U}}, only {{mvar|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.
5. * Every $C^k(K)$ contains the constant {{math|0}} map, even if $K\; =\; \backslash varnothing.$
6. Let $C\_c^k(U)$ denote the set of all $f\; \backslash in\; C^k(U)$ such that $f\; \backslash in\; C^k(K)$ for some compact subset {{mvar|K}} of {{mvar|U}}.
7. * Equivalently, $C\_c^k(U)$ is the set of all $f\; \backslash in\; C^k(U)$ such that $f$ has compact support.
8. * $C\_c^k(U)$ is equal to the union of all $C^k(K)$ as $K$ ranges over $\backslash mathbb\{K\}.$
9. * If $f$ is a real-valued function on {{mvar|U}}, then $f$ is an element of $C\_c^k(U)$ if and only if $f$ is a $C^k$ bump function. Every real-valued test function on $U$ is always also a complex-valued test function on $U.$

}}

File:Bump.png $(x,y)\; \backslash in\; \backslash mathbf\{R\}^2\; \backslash mapsto\; \backslash Psi(r),$ where $r\; =\; (x^2\; +\; y^2)^\backslash frac\{1\}\{2\}$ and This function is a test function on $\backslash R^2$ and is an element of $C^\backslash infty\_c\backslash left(\backslash R^2\backslash right).$ The support of this function is the closed unit disk in $\backslash R^2.$ It is non-zero on the open unit disk and it is equal to {{math|0}} everywhere outside of it.]]

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

$$\backslash begin\{align\}$$

C^k(K) &\subseteq C^k_c(U) \subseteq C^k(U) \\

C^k(K) &\subseteq C^k(L) && \text{ if } K \subseteq L \\

C^k(K) &\subseteq C^j(K) && \text{ if } j \leq k \\

C_c^k(U) &\subseteq C^j_c(U) && \text{ if } j \leq k \\

C^k(U) &\subseteq C^j(U) && \text{ if } j \leq k \\

\end{align}

{{block indent|em=1.5|text=Definition: Elements of $C\_c^\backslash infty(U)$ are called {{em|test functions}} on {{mvar|U}} and $C\_c^\backslash infty(U)$ is called the {{em|space of test function}} on {{mvar|U}}. We will use both $\backslash mathcal\{D\}(U)$ and $C\_c^\backslash infty(U)$ to denote this space.}}

Distributions on {{mvar|U}} are defined to be the continuous linear functionals on $C\_c^\backslash infty(U)$ when this vector space is endowed with a particular topology called the {{em|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 {{mvar|T}} is a linear functional on $C\_c^\backslash infty(U)$ then the {{mvar|T}} is a distribution if and only if the following equivalent conditions are satisfied:

1. For every compact subset $K\backslash subseteq\; U$ there exist constants $C>0$ and $N\backslash in\; \backslash N$ (dependent on $K$) such that for all $f\; \backslash in\; C^\backslash infty(K),${{sfn|Trèves|2006|pp=222-223}} $$|T(f)|\; \backslash leq\; C\; \backslash sup\; \backslash \{|\backslash partial^\backslash alpha\; f(x)|:\; x\; \backslash in\; U,\; |\backslash alpha|\; \backslash leq\; N\backslash \}.$$
2. For every compact subset $K\backslash subseteq\; U$ there exist constants $C>0$ and $N\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C\_c^\backslash infty(U)$ with support contained in $K,$See for example {{harvnb|Grubb|2009|page=14}}. $$|T(f)|\; \backslash leq\; C\; \backslash sup\; \backslash \{|\backslash partial^\backslash alpha\; f(x)|:\; x\; \backslash in\; K,\; |\backslash alpha|\backslash leq\; N\backslash \}.$$
3. For any compact subset $K\backslash subseteq\; U$ and any sequence $\backslash \{f\_i\backslash \}\_\{i=1\}^\backslash infty$ in $C^\backslash infty(K),$ if $\backslash \{\backslash partial^\backslash alpha\; f\_i\backslash \}\_\{i=1\}^\backslash infty$ converges uniformly to zero on $K$ for all multi-indices $\backslash alpha$, then $T(f\_i)\; \backslash to\; 0.$

The above characterizations can be used to determine whether or not a linear functional is a distribution, but more advanced uses of distributions and test functions (such as applications to differential equations) is limited if no topologies are placed on $C\_c^\backslash infty(U)$ and $\backslash mathcal\{D\}(U).$

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 $C^\backslash infty(U)$ will be defined, then every $C^\backslash infty(K)$ will be endowed with the subspace topology induced on it by $C^\backslash infty(U),$ and finally the (non-metrizable) canonical LF-topology on $C\_c^\backslash infty(U)$ will be defined.

The space of distributions, being defined as the continuous dual space of $C\_c^\backslash infty(U),$ is then endowed with the (non-metrizable) strong dual topology induced by $C\_c^\backslash infty(U)$ 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 {{em|and}} nets), various (sub)spaces of distributions, and operations on distributions, including extending differential equations to distributions.

Throughout, $\backslash mathbb\{K\}$ will be any collection of compact subsets of $U$ such that (1) $U\; =\; \backslash bigcup\_\{K\; \backslash in\; \backslash mathbb\{K\}\}\; K,$ and (2) for any compact $K\_1,K\_2\; \backslash subseteq\; U$ there exists some $K\; \backslash in\; \backslash mathbb\{K\}$ such that $K\_1\backslash cup\; K\_2\; \backslash subseteq\; K.$ The most common choices for $\backslash mathbb\{K\}$ are:

• The set of all compact subsets of $U,$ or
• A set $\backslash left\backslash \{\backslash overline\{U\}\_1,\; \backslash overline\{U\}\_2,\; \backslash ldots\; \backslash right\backslash \}$ where $U\; =\; \backslash bigcup\_\{i=1\}^\backslash infty\; U\_i,$ and for all {{mvar|i}}, $\backslash overline\{U\}\_i\; \backslash subseteq\; U\_\{i+1\}$ and $U\_i$ is a relatively compact non-empty open subset of $U$ (here, "relatively compact" means that the closure of $U\_i,$ in either {{mvar|U}} or $\backslash R^n,$ is compact).

We make $\backslash mathbb\{K\}$ into a directed set by defining $K\_1\; \backslash leq\; K\_2$ if and only if $K\_1\; \backslash subseteq\; K\_2.$ Note that although the definitions of the subsequently defined topologies explicitly reference $\backslash mathbb\{K\},$ in reality they do not depend on the choice of $\backslash mathbb\{K\};$ that is, if $\backslash mathbb\{K\}\_1$ and $\backslash mathbb\{K\}\_2$ are any two such collections of compact subsets of $U,$ then the topologies defined on $C^k(U)$ and $C\_c^k(U)$ by using $\backslash mathbb\{K\}\_1$ in place of $\backslash mathbb\{K\}$ are the same as those defined by using $\backslash mathbb\{K\}\_2$ in place of $\backslash mathbb\{K\}.$

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.

{{block indent|em=1.5|text=Suppose $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \}$ and $K$ is an arbitrary compact subset of $U.$ Suppose $i$ an integer such that $0\; \backslash leq\; i\; \backslash leq\; k$Note that $i$ being an integer implies $i\; \backslash neq\; \backslash infty.$ This is sometimes expressed as Since $\backslash infty\; +\; 1\; =\; \backslash infty,$ the inequality "" means: if $k\; =\; \backslash infty,$ while if $k\; \backslash neq\; \backslash infty$ then it means $0\; \backslash leq\; i\; \backslash leq\; k.$ and $p$ is a multi-index with length $|\; p|\backslash leq\; k.$ For $K\; \backslash neq\; \backslash varnothing,$ define:

$$\backslash begin\{alignat\}\{4\}$$

\text{ (1) }\ & s_{p,K}(f) &&:= \sup_{x_0 \in K} \left| \partial^p f(x_0) \right| \\[4pt]

\text{ (2) }\ & q_{i,K}(f) &&:= \sup_{|p| \leq i} \left(\sup_{x_0 \in K} \left| \partial^p f(x_0) \right|\right) = \sup_{|p| \leq i} \left(s_{p, K}(f)\right) \\[4pt]

\text{ (3) }\ & r_{i,K}(f) &&:= \sup_{\stackrel{|p| \leq i}{x_0 \in K}} \left| \partial^p f(x_0) \right| \\[4pt]

\text{ (4) }\ & t_{i,K}(f) &&:= \sup_{x_0 \in K} \left(\sum_{|p| \leq i} \left| \partial^p f(x_0) \right|\right)

\end{alignat}

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

}}

All of the functions above are non-negative $\backslash R$-valuedThe image of the compact set $K$ under a continuous $\backslash R$-valued map (for example, $x\; \backslash mapsto\; \backslash left|\backslash partial^p\; f(x)\backslash right|$ for $x\; \backslash in\; U$) is itself a compact, and thus bounded, subset of $\backslash R.$ If $K\; \backslash neq\; \backslash varnothing$ then this implies that each of the functions defined above is $\backslash R$-valued (that is, none of the supremums above are ever equal to $\backslash infty$). 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

$$\backslash begin\{alignat\}\{4\}$$

A ~:= \quad &\{q_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\

B ~:= \quad &\{r_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\

C ~:= \quad &\{t_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\

D ~:= \quad &\{s_{p,K} &&: \;K \text{ compact and } \;&&p \in \N^n \text{ satisfies } \;&&|p| \leq k\}

\end{alignat}

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 $C$).

{{block indent|em=1.5|text=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 {{em|all}} of the seminorms in $A\; \backslash cup\; B\; \backslash cup\; C\; \backslash cup\; D.$}}

With this topology, $C^k(U)$ becomes a locally convex Fréchet space that is {{em|not}} normable. Every element of $A\; \backslash cup\; B\; \backslash cup\; C\; \backslash cup\; D$ is a continuous seminorm on $C^k(U).$

Under this topology, a net $(f\_i)\_\{i\backslash in\; I\}$ in $C^k(U)$ converges to $f\; \backslash in\; C^k(U)$ if and only if for every multi-index $p$ with and every compact $K,$ the net of partial derivatives $\backslash left(\backslash partial^p\; f\_i\backslash right)\_\{i\; \backslash in\; I\}$ converges uniformly to $\backslash partial^p\; f$ on $K.${{sfn|Trèves|2006|pp=85-89}} For any $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \},$ any (von Neumann) bounded subset of $C^\{k+1\}(U)$ is a relatively compact subset of $C^k(U).${{sfn|Trèves|2006|pp=142-149}} In particular, a subset of $C^\backslash infty(U)$ is bounded if and only if it is bounded in $C^i(U)$ for all $i\; \backslash in\; \backslash N.${{sfn|Trèves|2006| pp=142-149}} The space $C^k(U)$ is a Montel space if and only if $k\; =\; \backslash infty.${{sfn|Trèves|2006|pp=356-358}}

The topology on $C^\backslash infty(U)$ is the superior limit of the subspace topologies induced on $C^\backslash infty(U)$ by the TVSs $C^i(U)$ as {{mvar|i}} ranges over the non-negative integers.{{sfn|Trèves|2006|pp=85-89}} A subset $W$ of $C^\backslash infty(U)$ is open in this topology if and only if there exists $i\backslash in\; \backslash N$ such that $W$ is open when $C^\backslash infty(U)$ is endowed with the subspace topology induced on it by $C^i(U).$

If the family of compact sets $\backslash mathbb\{K\}\; =\; \backslash left\backslash \{\backslash overline\{U\}\_1,\; \backslash overline\{U\}\_2,\; \backslash ldots\; \backslash right\backslash \}$ satisfies $U\; =\; \backslash bigcup\_\{j=1\}^\backslash infty\; U\_j$ and $\backslash overline\{U\}\_i\; \backslash subseteq\; U\_\{i+1\}$ for all $i,$ then a complete translation-invariant metric on $C^\backslash infty(U)$ 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 $(r\_\{i,K\_i\})\_\{i=1\}^\backslash infty$ results in the metric $$d(f,\; g)\; :=\; \backslash sum^\backslash infty\_\{i=1\}\; \backslash frac\{1\}\{2^i\}\; \backslash frac\{r\_\{i,\; \backslash overline\{U\}\_i\}(f\; -\; g)\}\{1\; +\; r\_\{i,\; \backslash overline\{U\}\_i\}(f\; -\; g)\}\; =\; \backslash sum^\backslash infty\_\{i=1\}\; \backslash frac\{1\}\{2^i\}\; \backslash frac\{\backslash sup\_$$

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

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

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

For any compact subset $K\; \backslash subseteq\; U,$ $C^k(K)$ is a closed subspace of the Fréchet space $C^k(U)$ and is thus also a Fréchet space. For all compact $K,\; L\; \backslash subseteq\; U$ satisfying $K\; \backslash subseteq\; L,$ denote the inclusion map by $\backslash operatorname\{In\}\_K^L\; :\; C^k(K)\; \backslash to\; C^k(L).$ 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 $C^k(K)$ is identical to the subspace topology it inherits from $C^k(L),$ and also $C^k(K)$ is a closed subset of $C^k(L).$ The interior of $C^\backslash infty(K)$ relative to $C^\backslash infty(U)$ is empty.{{sfn|Rudin|1991|pp=149-181}}

If $k$ is finite then $C^k(K)$ is a Banach space{{sfn|Trèves|2006|pp=131-134}} with a topology that can be defined by the norm

And when $k\; =\; 2,$ then $\backslash ,C^k(K)$ is even a Hilbert space.{{sfn|Trèves|2006|pp=131-134}} The space $C^\backslash infty(K)$ is a distinguished Schwartz Montel space so if $C^\backslash infty(K)\; \backslash neq\; \backslash \{0\backslash \}$ then it is {{em|not}} normable and thus {{em|not}} a Banach space (although like all other $C^k(K),$ it is a Fréchet space).

{{anchor|Omitting the open set from notation}}

The definition of $C^k(K)$ depends on {{mvar|U}} so we will let $C^k(K;U)$ denote the topological space $C^k(K),$ which by definition is a topological subspace of $C^k(U).$ Suppose $V$ is an open subset of $\backslash R^n$ containing $U$ and for any compact subset $K\; \backslash subseteq\; V,$ let $C^k(K;\; V)$ is the vector subspace of $C^k(V)$ consisting of maps with support contained in $K.$ Given $f\; \backslash in\; C\_c^k(U),$ its {{em|trivial extension to {{mvar|V}}}} is by definition, the function $I(f)\; :=\; F\; :\; V\; \backslash to\; \backslash Complex$ defined by:

so that $F\; \backslash in\; C^k(V).$ Let $I\; :\; C\_c^k(U)\; \backslash to\; C^k(V)$ denote the map that sends a function in $C\_c^k(U)$ to its trivial extension on {{mvar|V}}. This map is a linear injection and for every compact subset $K\; \backslash subseteq\; U$ (where $K$ is also a compact subset of $V$ since $K\; \backslash subseteq\; U\; \backslash subseteq\; V$) we have

$$\backslash begin\{alignat\}\{4\}$$

I\left(C^k(K; U)\right) &~=~ C^k(K; V) \qquad \text{ and thus } \\

I\left(C_c^k(U)\right) &~\subseteq~ C_c^k(V)

\end{alignat}

If {{mvar|I}} is restricted to $C^k(K;\; U)$ then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism):

$$\backslash begin\{alignat\}\{4\}$$

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

& f && \mapsto\,&& I(f) \\

\end{alignat}

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

$$C^k(K;\; U)\; \backslash to\; C^k(V),\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; C^k(K;\; U)\; \backslash to\; C\_c^k(V),$$

(the topology on $C\_c^k(V)$ is the canonical LF topology, which is defined later).

Using the injection

$$I\; :\; C\_c^k(U)\; \backslash to\; C^k(V)$$

the vector space $C\_c^k(U)$ is canonically identified with its image in $C\_c^k(V)\; \backslash subseteq\; C^k(V)$ (however, if $U\; \backslash neq\; V$ then $I\; :\; C\_c^\backslash infty(U)\backslash to\; C\_c^\backslash infty(V)$ is {{em|not}} a topological embedding when these spaces are endowed with their canonical LF topologies, although it is continuous).{{sfn|Trèves|2006|pp=245-247}}

Because $C^k(K;\; U)\; \backslash subseteq\; C\_c^k(U),$ through this identification, $C^k(K;\; U)$ can also be considered as a subset of $C^k(V).$ Importantly, the subspace topology $C^k(K;\; U)$ inherits from $C^k(U)$ (when it is viewed as a subset of $C^k(U)$) is identical to the subspace topology that it inherits from $C^k(V)$ (when $C^k(K;\; U)$ is viewed instead as a subset of $C^k(V)$ via the identification). Thus the topology on $C^k(K;U)$ is independent of the open subset {{mvar|U}} of $\backslash R^n$ that contains {{mvar|K}}.{{sfn|Rudin|1991|pp=149-181}} This justifies the practice of writing $C^k(K)$ instead of $C^k(K;\; U).$

{{See also|LF-space|Topology of uniform convergence}}

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

{{block indent|em=1.5|text=$C\_c^\backslash infty(U)$ is called the {{em|space of test functions on $U$}} and it may also be denoted by $\backslash mathcal\{D\}(U).$}}

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 {{mvar|K}} and {{mvar|L}}, we declare that $K\; \backslash leq\; L$ if and only if $K\; \backslash subseteq\; L,$ which in particular makes the collection $\backslash mathbb\{K\}$ of compact subsets of {{mvar|U}} into a directed set (we say that such a collection is {{em|directed by subset inclusion}}). For all compact $K,\; L\; \backslash subseteq\; U$ satisfying $K\; \backslash subseteq\; L,$ there are inclusion maps

$$\backslash operatorname\{In\}\_K^L\; :\; C^k(K)\; \backslash to\; C^k(L)\backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash operatorname\{In\}\_K^U\; :\; C^k(K)\; \backslash to\; C\_c^k(U).$$

Recall from above that the map $\backslash operatorname\{In\}\_K^L\; :\; C^k(K)\; \backslash to\; C^k(L)$ is a topological embedding. The collection of maps $$\backslash left\backslash \{\backslash operatorname\{In\}\_K^L\; \backslash ;:\backslash ;\; K,\; L\; \backslash in\; \backslash mathbb\{K\}\; \backslash ;\backslash text\{\; and\; \}\backslash ;\; K\; \backslash subseteq\; L\; \backslash right\backslash \}$$

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

{{block indent|em=1.5|text=The {{em|canonical LF topology on $C\_c^k(U)$}} is the finest locally convex topology on $C\_c^k(U)$ making all of the inclusion maps $\backslash operatorname\{In\}\_K^U\; :\; C^k(K)\; \backslash to\; C\_c^k(U)$ continuous (where {{mvar|K}} ranges over {{nowrap|$\backslash mathbb\{K\}$).}}}}

{{block indent|em=1.5|text=As is common in mathematics literature, the space $C\_c^k(U)$ is henceforth assumed to be endowed with its canonical LF topology (unless explicitly stated otherwise).}}

If $\backslash cal\; U$ is a convex subset of $C\_c^k(U),$ then $\backslash cal\; U$ is a neighborhood of the origin in the canonical LF topology if and only if it satisfies the following condition:

{{NumBlk|:|For all $K\; \backslash in\; \backslash mathbb\{K\},$ $\{\backslash cal\; U\}\backslash cap\; C^k(K)$ is a neighborhood of the origin in $C^k(K).$|{{EquationRef|CN}}}}

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

A {{em|linear differential operator in {{mvar|U}} with smooth coefficients}} is a sum

$$P\; :=\; \backslash sum\_\{\backslash alpha\; \backslash in\; \backslash N^n\}\; c\_\backslash alpha\; \backslash partial^\backslash alpha$$

where $c\_\backslash alpha\; \backslash in\; C^\backslash infty(U)$ and all but finitely many of $c\_\backslash alpha$ are identically {{math|0}}. The integer $\backslash sup\; \backslash \{|\backslash alpha|:\; c\_\backslash alpha\; \backslash neq\; 0\backslash \}$ is called the {{em|order}} of the differential operator $P.$ If $P$ is a linear differential operator of order {{mvar|k}} then it induces a canonical linear map $C^k(U)\; \backslash to\; C^0(U)$ defined by $\backslash phi\; \backslash mapsto\; P\backslash phi,$ where we shall reuse notation and also denote this map by $P.${{sfn|Trèves|2006|pp=247-252}}

For any $1\; \backslash leq\; k\; \backslash leq\; \backslash infty,$ the canonical LF topology on $C\_c^k(U)$ is the weakest locally convex TVS topology making all linear differential operators in $U$ of order into continuous maps from $C\_c^k(U)$ into $C\_c^0(U).${{sfn|Trèves|2006|pp=247-252}}

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 $\backslash mathbb\{K\}$ of compact sets. And by considering different collections $\backslash mathbb\{K\}$ (in particular, those $\backslash mathbb\{K\}$ 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 $C\_c^k(U)$ into a Hausdorff locally convex strict LF-space (and also a strict LB-space if $k\; \backslash neq\; \backslash infty$), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).If we take $\backslash mathbb\{K\}$ to be the set of {{em|all}} compact subsets of {{mvar|U}} then we can use the universal property of direct limits to conclude that the inclusion $\backslash operatorname\{In\}\_K^U\; :\; C^k(K)\; \backslash to\; C\_c^k(U)$ is a continuous and even that they are topological embedding for every compact subset $K\; \backslash subseteq\; U.$ If however, we take $\backslash mathbb\{K\}$ to be the set of closures of some countable increasing sequence of relatively compact open subsets of {{mvar|U}} having all of the properties mentioned earlier in this in this article then we immediately deduce that $C\_c^k(U)$ is a Hausdorff locally convex strict LF-space (and even a strict LB-space when $k\; \backslash neq\; \backslash infty$). All of these facts can also be proved directly without using direct systems (although with more work).

From the universal property of direct limits, we know that if $u\; :\; C\_c^k(U)\; \backslash to\; Y$ is a linear map into a locally convex space {{mvar|Y}} (not necessarily Hausdorff), then {{mvar|u}} is continuous if and only if {{mvar|u}} is bounded if and only if for every $K\; \backslash in\; \backslash mathbb\{K\},$ the restriction of {{mvar|u}} to $C^k(K)$ is continuous (or bounded).{{sfn|Trèves|2006|pp=126-134}}{{sfn|Trèves|2006|pp=136-148}}

Suppose {{mvar|V}} is an open subset of $\backslash R^n$ containing $U.$ Let $I\; :\; C\_c^k(U)\backslash to\; C\_c^k(V)$ denote the map that sends a function in $C\_c^k(U)$ to its trivial extension on {{mvar|V}} (which was defined above). This map is a continuous linear map.{{sfn|Trèves|2006|pp=245-247}} If (and only if) $U\; \backslash neq\; V$ then $I\backslash left(C\_c^\backslash infty(U)\backslash right)$ is {{em|not}} a dense subset of $C\_c^\backslash infty(V)$ and $I\; :\; C\_c^\backslash infty(U)\backslash to\; C\_c^\backslash infty(V)$ is {{em|not}} a topological embedding.{{sfn|Trèves|2006|pp=245-247}} Consequently, if $U\; \backslash neq\; V$ then the transpose of $I\; :\; C\_c^\backslash infty(U)\backslash to\; C\_c^\backslash infty(V)$ is neither one-to-one nor onto.{{sfn|Trèves|2006|pp=245-247}}

A subset $B\; \backslash subseteq\; C\_c^k(U)$ is bounded in $C\_c^k(U)$ if and only if there exists some $K\; \backslash in\; \backslash mathbb\{K\}$ such that $B\; \backslash subseteq\; C^k(K)$ and $B$ is a bounded subset of $C^k(K).${{sfn|Trèves|2006|pp=136-148}} Moreover, if $K\; \backslash subseteq\; U$ is compact and $S\; \backslash subseteq\; C^k(K)$ then $S$ is bounded in $C^k(K)$ if and only if it is bounded in $C^k(U).$ For any $0\; \backslash leq\; k\; \backslash leq\; \backslash infty,$ any bounded subset of $C\_c^\{k+1\}(U)$ (resp. $C^\{k+1\}(U)$) is a relatively compact subset of $C\_c^k(U)$ (resp. $C^k(U)$), where $\backslash infty\; +\; 1\; =\; \backslash infty.${{sfn|Trèves|2006|pp=136-148}}

For all compact $K\; \backslash subseteq\; U,$ the interior of $C^k(K)$ in $C\_c^k(U)$ is empty so that $C\_c^k(U)$ is of the first category in itself. It follows from Baire's theorem that $C\_c^k(U)$ is {{em|not}} metrizable and thus also {{em|not}} normable (see this footnoteFor any TVS {{mvar|X}} (metrizable or otherwise), the notion of completeness depends entirely on a certain so-called "canonical uniformity" that is defined using {{em|only}} the subtraction operation (see the article Complete topological vector space for more details). In this way, the notion of a complete TVS does not {{em|require}} the existence of any metric. However, if the TVS {{mvar|X}} is metrizable and if $d$ is {{em|any}} translation-invariant metric on {{mvar|X}} that defines its topology, then {{mvar|X}} is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if $(X,\; d)$ is a complete metric space. So if a TVS {{mvar|X}} happens to have a topology that can be defined by such a metric {{mvar|d}} then {{mvar|d}} may be used to deduce the completeness of {{mvar|X}} but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the Cartesian product of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS $\backslash R^\backslash N,$ which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on $\backslash R^\backslash N$). for an explanation of how the non-metrizable space $C\_c^k(U)$ can be complete even though it does not admit a metric). The fact that $C\_c^\backslash infty(U)$ is a nuclear Montel space makes up for the non-metrizability of $C\_c^\backslash infty(U)$ (see this footnote for a more detailed explanation).One reason for giving $C\_c^\backslash infty(U)$ the canonical LF topology is because it is with this topology that $C\_c^\backslash infty(U)$ and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of topological vector spaces (TVSs) that are particularly similar to finite-dimensional Euclidean spaces: the Banach spaces (especially Hilbert spaces) and the nuclear Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are nuclear Montel Hilbert spaces but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. {{em|infinite}} differentiability, such as $C\_c^\backslash infty(U)$ and $C^\backslash infty(U)$) end up being nuclear TVSs while TVSs associated with {{em|finite}} continuous differentiability (such as $C^k(K)$ with {{mvar|K}} compact and $k\; \backslash neq\; \backslash infty$) often end up being non-nuclear spaces, such as Banach spaces.

Using the universal property of direct limits and the fact that the natural inclusions $\backslash operatorname\{In\}\_K^L\; :\; C^k(K)\; \backslash to\; C^k(L)$ are all topological embedding, one may show that all of the maps $\backslash operatorname\{In\}\_K^U\; :\; C^k(K)\; \backslash to\; C\_c^k(U)$ are also topological embeddings. Said differently, the topology on $C^k(K)$ is identical to the subspace topology that it inherits from $C\_c^k(U),$ where recall that $C^k(K)$'s topology was {{em|defined}} to be the subspace topology induced on it by $C^k(U).$ In particular, both $C\_c^k(U)$ and $C^k(U)$ induces the same subspace topology on $C^k(K).$ However, this does {{em|not}} imply that the canonical LF topology on $C\_c^k(U)$ is equal to the subspace topology induced on $C\_c^k(U)$ by $C^k(U)$; these two topologies on $C\_c^k(U)$ are in fact {{em|never}} equal to each other since the canonical LF topology is {{em|never}} metrizable while the subspace topology induced on it by $C^k(U)$ is metrizable (since recall that $C^k(U)$ is metrizable). The canonical LF topology on $C\_c^k(U)$ is actually {{em|strictly finer}} than the subspace topology that it inherits from $C^k(U)$ (thus the natural inclusion $C\_c^k(U)\backslash to\; C^k(U)$ is continuous but {{em|not}} a topological embedding).{{sfn|Trèves|2006|pp=131-134}}

Indeed, the canonical LF topology is so fine that if $C\_c^\backslash infty(U)\backslash to\; X$ denotes some linear map that is a "natural inclusion" (such as $C\_c^\backslash infty(U)\backslash to\; C^k(U),$ or $C\_c^\backslash infty(U)\backslash to\; L^p(U),$ 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 $C\_c^\backslash infty(U),$ the fine nature of the canonical LF topology means that more linear functionals on $C\_c^\backslash infty(U)$ end up being continuous ("more" means as compared to a coarser topology that we could have placed on $C\_c^\backslash infty(U)$ such as for instance, the subspace topology induced by some $C^k(U),$ which although it would have made $C\_c^\backslash infty(U)$ metrizable, it would have also resulted in fewer linear functionals on $C\_c^\backslash infty(U)$ being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making $C\_c^\backslash infty(U)$ into a complete TVS{{sfn|Rudin|1991|pp=149-155}}).

• The differentiation map $C\_c^\backslash infty(U)\; \backslash to\; C\_c^\backslash infty(U)$ is a continuous linear operator.{{sfn|Narici|Beckenstein|2011|pp=446-447}}
• The bilinear multiplication map $C^\backslash infty(\backslash R^m)\; \backslash times\; C\_c^\backslash infty(\backslash R^n)\; \backslash to\; C\_c^\backslash infty(\backslash R^\{m+n\})$ given by $(f,g)\backslash mapsto\; fg$ is {{em|not}} continuous; it is however, hypocontinuous.{{sfn|Trèves|2006|p=423}}

{{See also|Continuous linear functional}}

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

{{block indent|em=1.5|text=By definition, a {{em|distribution on {{mvar|U}}}} is defined to be a continuous linear functional on $C\_c^\backslash infty(U).$ Said differently, a distribution on {{mvar|U}} is an element of the continuous dual space of $C\_c^\backslash infty(U)$ when $C\_c^\backslash infty(U)$ is endowed with its canonical LF topology.}}

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

$$\backslash begin\{cases\}\; \backslash mathcal\{D\}^\{\backslash prime\}(U)\; \backslash times\; C\_c^\backslash infty(U)\; \backslash to\; \backslash R\; \backslash \backslash \; (T,\; f)\; \backslash mapsto\; \backslash langle\; T,\; f\; \backslash rangle\; :=\; T(f)\; \backslash end\{cases\}$$

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

Proposition. If {{mvar|T}} is a linear functional on $C\_c^\backslash infty(U)$ then the following are equivalent:

1. {{mvar|T}} is a distribution;
2. {{em|Definition}} : {{mvar|T}} is a continuous function.
3. {{mvar|T}} is continuous at the origin.
4. {{mvar|T}} is uniformly continuous.
5. {{mvar|T}} is a bounded operator.
6. {{mvar|T}} is sequentially continuous.
7. * explicitly, for every sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U)$ that converges in $C\_c^\backslash infty(U)$ to some $f\; \backslash in\; C\_c^\backslash infty(U),$ $\backslash lim\_\{i\; \backslash to\; \backslash infty\}\; T\backslash left(f\_i\backslash right)\; =\; T(f);$Even though the topology of $C\_c^\backslash infty(U)$ is not metrizable, a linear functional on $C\_c^\backslash infty(U)$ is continuous if and only if it is sequentially continuous.
8. {{mvar|T}} is sequentially continuous at the origin; in other words, {{mvar|T}} maps null sequences to null sequences.
9. * explicitly, for every sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U)$ that converges in $C\_c^\backslash infty(U)$ to the origin (such a sequence is called a {{em|null sequence}}), $\backslash lim\_\{i\; \backslash to\; \backslash infty\}\; T\backslash left(f\_i\backslash right)\; =\; 0.$
10. * a {{em|null sequence}} is by definition a sequence that converges to the origin.
11. {{mvar|T}} maps null sequences to bounded subsets.
12. * explicitly, for every sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U)$ that converges in $C\_c^\backslash infty(U)$ to the origin, the sequence $\backslash left(T\backslash left(f\_i\backslash right)\backslash right)\_\{i=1\}^\backslash infty$ is bounded.
13. {{mvar|T}} maps Mackey convergent null sequences to bounded subsets;
14. * explicitly, for every Mackey convergent null sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U),$ the sequence $\backslash left(T\backslash left(f\_i\backslash right)\backslash right)\_\{i=1\}^\backslash infty$ is bounded.
15. * a sequence $f\_\{\backslash bull\}\; =\; \backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ is said to be {{em|Mackey convergent to {{math|0}}}} if there exists a divergent sequence $r\_\{\backslash bull\}\; =\; \backslash left(r\_i\backslash right)\_\{i=1\}^\backslash infty\; \backslash to\; \backslash infty$ of positive real number such that the sequence $\backslash left(r\_i\; f\_i\backslash right)\_\{i=1\}^\backslash infty$ is bounded; every sequence that is Mackey convergent to {{math|0}} necessarily converges to the origin (in the usual sense).
16. The kernel of {{mvar|T}} is a closed subspace of $C\_c^\backslash infty(U).$
17. The graph of {{mvar|T}} is closed.

1. There exists a continuous seminorm $g$ on $C\_c^\backslash infty(U)$ such that $|T|\; \backslash leq\; g.$
2. There exists a constant $C\; >\; 0,$ a collection of continuous seminorms, $\backslash mathcal\{P\},$ that defines the canonical LF topology of $C\_c^\backslash infty(U),$ and a finite subset $\backslash left\backslash \{g\_1,\; \backslash ldots,\; g\_m\backslash right\backslash \}\; \backslash subseteq\; \backslash mathcal\{P\}$ such that $|T|\; \backslash leq\; C(g\_1\; +\; \backslash cdots\; g\_m);$If $\backslash mathcal\{P\}$ is also a directed set under the usual function comparison then we can take the finite collection to consist of a single element.
3. For every compact subset $K\; \backslash subseteq\; U$ there exist constants $C\; >\; 0$ and $N\; \backslash in\; \backslash N$ such that for all $f\; \backslash in\; C^\backslash infty(K),${{sfn|Trèves|2006|pp=222-223}} $$|T(f)|\; \backslash leq\; C\; \backslash sup\; \backslash \{|\backslash partial^p\; f(x)|\; :\; x\; \backslash in\; U,\; |\backslash alpha|\backslash leq\; N\backslash \}.$$
4. For every compact subset $K\backslash subseteq\; U$ there exist constants $C\_K>0$ and $N\_K\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C\_c^\backslash infty(U)$ with support contained in $K,$ $$|T(f)|\; \backslash leq\; C\_K\; \backslash sup\; \backslash \{\backslash left|\backslash partial^\backslash alpha\; f(x)\backslash right|\; :\; x\; \backslash in\; K,\; |\backslash alpha|\backslash leq\; N\_K\backslash \}.$$
5. For any compact subset $K\backslash subseteq\; U$ and any sequence $\backslash \{f\_i\backslash \}\_\{i=1\}^\backslash infty$ in $C^\backslash infty(K),$ if $\backslash \{\backslash partial^p\; f\_i\backslash \}\_\{i=1\}^\backslash infty$ converges uniformly to zero for all multi-indices $p,$ then $T(f\_i)\; \backslash to\; 0.$
6. Any of the {{em|three}} statements immediately above (that is, statements 14, 15, and 16) but with the additional requirement that compact set $K$ belongs to $\backslash mathbb\{K\}.$

{{See also|Strong dual space|Polar topology|Dual topology|Dual system}}

{{block indent|em=1.5|text=Definition and notation: {{em|The space of distributions on {{mvar|U}}}}, denoted by $\backslash mathcal\{D\}^\{\backslash prime\}(U),$ is the continuous dual space of $C\_c^\backslash infty(U)$ endowed with the topology of uniform convergence on bounded subsets of $C\_c^\backslash infty(U).${{sfn|Trèves|2006|pp=131-134}} More succinctly, the space of distributions on {{mvar|U}} is $\backslash mathcal\{D\}^\{\backslash prime\}(U)\; :=\; \backslash left(C\_c^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b.$}}

The topology of uniform convergence on bounded subsets is also called {{em|the strong dual topology}}.In functional analysis, the strong dual topology is often the "standard" or "default" topology placed on the continuous dual space $X\text{'},$ where if {{mvar|X}} is a normed space then this strong dual topology is the same as the usual norm-induced topology on $X\text{'}.$ This topology is chosen because it is with this topology that $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.See for example {{harvnb|Schaefer|Wolff|1999|p=173}}. No matter what dual topology is placed on $\backslash mathcal\{D\}^\{\backslash prime\}(U),$Technically, the topology must be coarser than the strong dual topology and also simultaneously be finer that the weak* topology. a {{em|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, $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ will be a non-metrizable, locally convex topological vector space. The space $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ is separable and has the strong Pytkeev propertyGabriyelyan, S.S. Kakol J., and·Leiderman, A. [https://prometeo2013058.blogs.upv.es/files/2014/02/sPp-MM.pdf "The strong Pitkeev property for topological groups and topological vector spaces"] but it is neither a k-space nor a sequential space, which in particular implies that it is not metrizable and also that its topology can {{em|not}} be defined using only sequences.

The canonical LF topology makes $C\_c^k(U)$ into a complete distinguished strict LF-space (and a strict LB-space if and only if $k\; \backslash neq\; \backslash infty${{sfn|Trèves|2006|pp=195-201}}), which implies that $C\_c^k(U)$ is a meager subset of itself.{{sfn|Narici|Beckenstein|2011|p=435}} Furthermore, $C\_c^k(U),$ as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of $C\_c^k(U)$ is a Fréchet space if and only if $k\; \backslash neq\; \backslash infty$ so in particular, the strong dual of $C\_c^\backslash infty(U),$ which is the space $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ of distributions on {{mvar|U}}, is {{em|not}} metrizable (note that the weak-* topology on $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives $\backslash mathcal\{D\}^\{\backslash prime\}(U)$).

The three spaces $C\_c^\backslash infty(U),$ $C^\backslash infty(U),$ and the Schwartz space $\backslash mathcal\{S\}(\backslash R^n),$ as well as the strong duals of each of these three spaces, are complete nuclear{{sfn|Trèves|2006|pp=526-534}} Montel{{sfn|Trèves|2006|p=357}} bornological spaces, which implies that all six of these locally convex spaces are also paracompact{{cite web |title=Topological vector space |author= |website=Encyclopedia of Mathematics |access-date=September 6, 2020 |url=https://encyclopediaofmath.org/wiki/Topological_vector_space |quote="It is a Montel space, hence paracompact, and so normal."}} reflexive barrelled Mackey spaces. The spaces $C^\backslash infty(U)$ and $\backslash mathcal\{S\}(\backslash R^n)$ are both distinguished Fréchet spaces. Moreover, both $C\_c^\backslash infty(U)$ and $\backslash mathcal\{S\}(\backslash R^n)$ are Schwartz TVSs.

The strong dual spaces of $C^\backslash infty(U)$ and $\backslash mathcal\{S\}(\backslash R^n)$ are sequential spaces but not Fréchet-Urysohn spaces. Moreover, neither the space of test functions $C\_c^\backslash infty(U)$ nor its strong dual $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ is a sequential space (not even an Ascoli space),Gabriyelyan, Saak [https://arxiv.org/pdf/1702.07867.pdf "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"] (2017)T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36. which in particular implies that their topologies can {{em|not}} be defined entirely in terms of convergent sequences.

A sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^k(U)$ converges in $C\_c^k(U)$ if and only if there exists some $K\; \backslash in\; \backslash mathbb\{K\}$ such that $C^k(K)$ contains this sequence and this sequence converges in $C^k(K)$; equivalently, it converges if and only if the following two conditions hold:According to {{harvnb|Gel'fand|Shilov|1966–1968|loc=v. 1, §1.2}}

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

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

Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology,{{sfn|Trèves|2006|pp=351-359}} which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually {{em|define}} the convergence of a sequence of distributions; this is fine for sequences but it does {{em|not}} extend to the convergence of nets of distributions since a net may converge pointwise but fail to converge in the strong dual topology).

Sequences characterize continuity of linear maps valued in locally convex space. Suppose {{mvar|X}} is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map $F\; :\; X\; \backslash to\; Y$ into a locally convex space {{mvar|Y}} is continuous if and only if it maps null sequencesA {{em|null sequence}} is a sequence that converges to the origin. in {{mvar|X}} to bounded subsets of {{mvar|Y}}.Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences. More generally, such a linear map $F\; :\; X\; \backslash to\; Y$ is continuous if and only if it maps Mackey convergent null sequencesA sequence $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\backslash infty$ is said to be {{em|Mackey convergent to {{math|0}} in $X,$}} if there exists a divergent sequence $r\_\{\backslash bull\}\; =\; \backslash left(r\_i\backslash right)\_\{i=1\}^\backslash infty\; \backslash to\; \backslash infty$ of positive real number such that $\backslash left(r\_i\; x\_i\backslash right)\_\{i=1\}^\backslash infty$ is a bounded set in $X.$ to bounded subsets of $Y.$ So in particular, if a linear map $F\; :\; X\; \backslash to\; Y$ into a locally convex space is sequentially continuous at the origin then it is continuous.{{sfn|Narici|Beckenstein|2011|pp=441-457}} However, this does {{em|not}} necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs.

For every $k\; \backslash in\; \backslash \{0,\; 1,\; \backslash ldots,\; \backslash infty\backslash \},\; C\_c^\backslash infty(U)$ is sequentially dense in $C\_c^k(U).${{sfn|Trèves|2006|pp=150-160}} Furthermore, $\backslash \{D\_\backslash phi\; :\; \backslash phi\; \backslash in\; C\_c^\backslash infty(U)\backslash \}$ is a sequentially dense subset of $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ (with its strong dual topology){{sfn|Trèves|2006|pp=300-304}} and also a sequentially dense subset of the strong dual space of $C^\backslash infty(U).${{sfn|Trèves|2006|pp=300-304}}

{{Main|Limit of distributions}}

A sequence of distributions $(T\_i)\_\{i=1\}^\backslash infty$ converges with respect to the weak-* topology on $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ to a distribution {{mvar|T}} if and only if

$$\backslash langle\; T\_i,\; f\; \backslash rangle\; \backslash to\; \backslash langle\; T,\; f\; \backslash rangle$$

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

and $T\_m$ is the distribution corresponding to $f\_m,$ then

$$\backslash langle\; T\_m,\; f\; \backslash rangle\; =\; m\; \backslash int\_0^\{\backslash frac\{1\}\{m\}\}\; f(x)\backslash ,\; dx\; \backslash to\; f(0)\; =\; \backslash langle\; \backslash delta,\; f\; \backslash rangle$$

as $m\; \backslash to\; \backslash infty,$ so $T\_m\; \backslash to\; \backslash delta$ in $\backslash mathcal\{D\}^\{\backslash prime\}(\backslash R).$ Thus, for large $m,$ the function $f\_m$ can be regarded as an approximation of the Dirac delta distribution.

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

{{anchor|Transpose of a linear operator}}

{{Main|Transpose of a linear map}}

Operations on distributions and spaces of distributions are often defined by means of the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well known in functional analysis.{{harvnb|Strichartz|1994|loc=§2.3}}; {{harvnb|Trèves|2006}}. For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space). In general the transpose of a continuous linear map $A\; :\; X\; \backslash to\; Y$ is the linear map

$$\{\}^\{t\}A\; :\; Y\text{'}\; \backslash to\; X\text{'}\; \backslash qquad\; \backslash text\{\; defined\; by\; \}\; \backslash qquad\; \{\}^\{t\}A(y\text{'})\; :=\; y\text{'}\; \backslash circ\; A,$$

or equivalently, it is the unique map satisfying $\backslash langle\; y\text{'},\; A(x)\backslash rangle\; =\; \backslash left\backslash langle\; \{\}^\{t\}A\; (y\text{'}),\; x\; \backslash right\backslash rangle$ for all $x\; \backslash in\; X$ and all $y\text{'}\; \backslash in\; Y\text{'}$ (the prime symbol in $y\text{'}$ does not denote a derivative of any kind; it merely indicates that $y\text{'}$ is an element of the continuous dual space $Y\text{'}$). Since $A$ is continuous, the transpose $\{\}^\{t\}A\; :\; Y\text{'}\; \backslash to\; X\text{'}$ is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).

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

$$\backslash langle\; \{\}^\{t\}A(T),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T,\; A(\backslash phi)\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(U)\; \backslash text\{\; and\; all\; \}\; T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U).$$

Since $\backslash mathcal\{D\}(U)$ is dense in $\backslash mathcal\{D\}\text{'}(U)$ (here, $\backslash mathcal\{D\}(U)$ actually refers to the set of distributions $\backslash left\backslash \{D\_\backslash psi\; :\; \backslash psi\; \backslash in\; \backslash mathcal\{D\}(U)\backslash right\backslash \}$) it is sufficient that the defining equality hold for all distributions of the form $T\; =\; D\_\backslash psi$ where $\backslash psi\; \backslash in\; \backslash mathcal\{D\}(U).$ Explicitly, this means that a continuous linear map $B\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)$ is equal to $\{\}^\{t\}A$ if and only if the condition below holds:

$$\backslash langle\; B(D\_\backslash psi),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; \{\}^\{t\}A(D\_\backslash psi),\; \backslash phi\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi,\; \backslash psi\; \backslash in\; \backslash mathcal\{D\}(U)$$

where the right hand side equals $\backslash langle\; \{\}^\{t\}A(D\_\backslash psi),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; D\_\backslash psi,\; A(\backslash phi)\; \backslash rangle\; =\; \backslash langle\; \backslash psi,\; A(\backslash phi)\; \backslash rangle\; =\; \backslash int\_U\; \backslash psi\; \backslash cdot\; A(\backslash phi)\; \backslash ,dx.$

Let $V\; \backslash subseteq\; U$ be open subsets of $\backslash R^n.$

Every function $f\; \backslash in\; \backslash mathcal\{D\}(V)$ can be {{em|extended by zero}} from its domain $V$ to a function on $U$ by setting it equal to $0$ on the complement $U\; \backslash setminus\; V.$ This extension is a smooth compactly supported function called the {{em|trivial extension of $f$ to $U$}} and it will be denoted by $E\_\{VU\}\; (f).$

This assignment $f\; \backslash mapsto\; E\_\{VU\}\; (f)$ defines the {{em|trivial extension}} operator

$E\_\{VU\}\; :\; \backslash mathcal\{D\}(V)\; \backslash to\; \backslash mathcal\{D\}(U),$

which is a continuous injective linear map. It is used to canonically identify $\backslash mathcal\{D\}(V)$ as a vector subspace of $\backslash mathcal\{D\}(U)$ (although {{em|not}} as a topological subspace).

Its transpose (explained here)

$$\backslash rho\_\{VU\}\; :=\; \{\}^\{t\}E\_\{VU\}\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(V),$$

is called the {{em|{{visible anchor|restriction map|text=restriction to $V$ of distributions in $U$}}}}{{sfn|Trèves|2006|pp=245-247}} and as the name suggests, the image $\backslash rho\_\{VU\}(T)$ of a distribution $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ under this map is a distribution on $V$ called the restriction of $T$ to $V.$ The defining condition of the restriction $\backslash rho\_\{VU\}(T)$ is:

$$\backslash langle\; \backslash rho\_\{VU\}\; T,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T,\; E\_\{VU\}\; \backslash phi\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(V).$$

If $V\; \backslash neq\; U$ then the (continuous injective linear) trivial extension map $E\_\{VU\}\; :\; \backslash mathcal\{D\}(V)\; \backslash to\; \backslash mathcal\{D\}(U)$ is {{em|not}} a topological embedding (in other words, if this linear injection was used to identify $\backslash mathcal\{D\}(V)$ as a subset of $\backslash mathcal\{D\}(U)$ then $\backslash mathcal\{D\}(V)$'s topology would strictly finer than the subspace topology that $\backslash mathcal\{D\}(U)$ induces on it; importantly, it would {{em|not}} be a topological subspace since that requires equality of topologies) and its range is also {{em|not}} dense in its codomain $\backslash mathcal\{D\}(U).${{sfn|Trèves|2006|pp=245-247}} Consequently, if $V\; \backslash neq\; U$ then the restriction mapping is neither injective nor surjective.{{sfn|Trèves|2006|pp=245-247}} A distribution $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(V)$ is said to be {{em|extendible to {{mvar|U}}}} if it belongs to the range of the transpose of $E\_\{VU\}$ and it is called {{em|extendible}} if it is extendable to $\backslash R^n.${{sfn|Trèves|2006|pp=245-247}}

Unless $U\; =\; V,$ the restriction to $V$ is neither injective nor surjective.

For all and all all of the following canonical injections are continuous and have an image/range that is a dense subset of their codomain:{{sfn|Trèves|2006|pp=131-135}}{{sfn|Trèves|2006|pp=240-245}}

$$\backslash begin\{matrix\}$$

C_c^\infty(U) & \to & C_c^k(U) & \to & C_c^0(U) & \to & L_c^\infty(U) & \to & L_c^{p+1}(U) & \to & L_c^p(U) & \to & L_c^1(U) \\

\downarrow & &\downarrow && \downarrow && && && && && \\

C^\infty(U) & \to & C^k(U) & \to & C^0(U) && && && && &&

\end{matrix}

where the topologies on the LB-spaces $L\_c^p(U)$ are the canonical LF topologies as defined below (so in particular, they are not the usual norm topologies).

The range of each of the maps above (and of any composition of the maps above) is dense in the codomain. Indeed, $C\_c^\backslash infty(U)$ is even sequentially dense in every $C\_c^k(U).${{sfn|Trèves|2006|pp=150-160}} For every $1\; \backslash leq\; p\; \backslash leq\; \backslash infty,$ the canonical inclusion $C\_c^\backslash infty(U)\; \backslash to\; L^p(U)$ into the normed space $L^p(U)$ (here $L^p(U)$ has its usual norm topology) is a continuous linear injection and the range of this injection is dense in its codomain if and only if $p\; \backslash neq\; \backslash infty$ .{{sfn|Trèves|2006|pp=240-245}}

Suppose that $X$ is one of the LF-spaces $C\_c^k(U)$ (for $k\; \backslash in\; \backslash \{0,\; 1,\; \backslash ldots,\; \backslash infty\backslash \}$) or LB-spaces $L^p\_c(U)$ (for $1\; \backslash leq\; p\; \backslash leq\; \backslash infty$) or normed spaces $L^p(U)$ (for ).{{sfn|Trèves|2006|pp=240-245}} Because the canonical injection $\backslash operatorname\{In\}\_X\; :\; C\_c^\backslash infty(U)\; \backslash to\; X$ is a continuous injection whose image is dense in the codomain, this map's transpose $\{\}^\{t\}\backslash operatorname\{In\}\_X\; :\; X\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)\; =\; \backslash left(C\_c^\backslash infty(U)\backslash right)\text{'}\_b$ is a continuous injection. This injective transpose map thus allows the continuous dual space $X\text{'}$ of $X$ to be identified with a certain vector subspace of the space $\backslash mathcal\{D\}\text{'}(U)$ of all distributions (specifically, it is identified with the image of this transpose map). This continuous transpose map is not necessarily a TVS-embedding so the topology that this map transfers from its domain to the image $\backslash operatorname\{Im\}\backslash left(\{\}^\{t\}\backslash operatorname\{In\}\_X\backslash right)$ is finer than the subspace topology that this space inherits from $\backslash mathcal\{D\}^\{\backslash prime\}(U).$

A linear subspace of $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ carrying a locally convex topology that is finer than the subspace topology induced by $\backslash mathcal\{D\}^\{\backslash prime\}(U)\; =\; \backslash left(C\_c^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b$ is called {{em|a space of distributions}}.{{sfn|Trèves|2006|pp=240-252}}

Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order $\backslash leq$ some integer, distributions induced by a positive Radon measure, distributions induced by an $L^p$-function, etc.) and any representation theorem about the dual space of {{mvar|X}} may, through the transpose $\{\}^\{t\}\backslash operatorname\{In\}\_X\; :\; X\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}^\{\backslash prime\}(U),$ be transferred directly to elements of the space $\backslash operatorname\{Im\}\; \backslash left(\{\}^\{t\}\backslash operatorname\{In\}\_X\backslash right).$

Given $1\; \backslash leq\; p\; \backslash leq\; \backslash infty,$ the vector space $L\_c^p(U)$ of {{visible anchor|compactly supported Lp space|text=compactly supported $L^p$ functions}} on $U$ and its topology are defined as direct limits of the spaces $L\_c^p(K)$ in a manner analogous to how the canonical LF-topologies on $C\_c^k(U)$ were defined.

For any compact $K\; \backslash subseteq\; U,$ let $L^p(K)$ denote the set of all element in $L^p(U)$ (which recall are equivalence class of Lebesgue measurable $L^p$ functions on $U$) having a representative $f$ whose support (which recall is the closure of $\backslash \{u\; \backslash in\; U\; :\; f(x)\; \backslash neq\; 0\backslash \}$ in $U$) is a subset of $K$ (such an $f$ is almost everywhere defined in $K$).

The set $L^p(K)$ is a closed vector subspace $L^p(U)$ and is thus a Banach space and when $p\; =\; 2,$ even a Hilbert space.{{sfn|Trèves|2006|pp=131-135}}

Let $L\_c^p(U)$ be the union of all $L^p(K)$ as $K\; \backslash subseteq\; U$ ranges over all compact subsets of $U.$

The set $L\_c^p(U)$ is a vector subspace of $L^p(U)$ whose elements are the (equivalence classes of) compactly supported $L^p$ functions defined on $U$ (or almost everywhere on $U$).

Endow $L\_c^p(U)$ with the final topology (direct limit topology) induced by the inclusion maps $L^p(K)\; \backslash to\; L\_c^p(U)$ as $K\; \backslash subseteq\; U$ ranges over all compact subsets of $U.$

This topology is called the {{em|canonical LF topology}} and it is equal to the final topology induced by any countable set of inclusion maps $L^p(K\_n)\; \backslash to\; L\_c^p(U)$ ($n\; =\; 1,\; 2,\; \backslash ldots$) where $K\_1\; \backslash subseteq\; K\_2\; \backslash subseteq\; \backslash cdots$ are any compact sets with union equal to $U.${{sfn|Trèves|2006|pp=131-135}}

This topology makes $L\_c^p(U)$ into an LB-space (and thus also an LF-space) with a topology that is strictly finer than the norm (subspace) topology that $L^p(U)$ induces on it.

The inclusion map $\backslash operatorname\{In\}\; :\; C\_c^\backslash infty(U)\; \backslash to\; C\_c^0(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $\{\}^\{t\}\backslash operatorname\{In\}\; :\; \backslash left(C\_c^0(U)\backslash right)^\{\backslash prime\}\_b\; \backslash to\; \backslash mathcal\{D\}^\{\backslash prime\}(U)\; =\; \backslash left(C\_c^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b$ is also a continuous injection.

Note that the continuous dual space $\backslash left(C\_c^0(U)\backslash right)^\{\backslash prime\}\_b$ can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals $T\; \backslash in\; \backslash left(C\_c^0(U)\backslash right)^\{\backslash prime\}\_b$ and integral with respect to a Radon measure; that is,

• if $T\; \backslash in\; \backslash left(C\_c^0(U)\backslash right)^\{\backslash prime\}\_b$ then there exists a Radon measure $\backslash mu$ on {{mvar|U}} such that for all $f\; \backslash in\; C\_c^0(U),\; T(f)\; =\; \backslash textstyle\; \backslash int\_U\; f\; \backslash ,\; d\backslash mu,$ and
• if $\backslash mu$ is a Radon measure on {{mvar|U}} then the linear functional on $C\_c^0(U)$ defined by $C\_c^0(U)\; \backslash ni\; f\; \backslash mapsto\; \backslash textstyle\; \backslash int\_U\; f\; \backslash ,\; d\backslash mu$ is continuous.

Through the injection $\{\}^\{t\}\backslash operatorname\{In\}\; :\; \backslash left(C\_c^0(U)\backslash right)^\{\backslash prime\}\_b\; \backslash to\; \backslash mathcal\{D\}^\{\backslash prime\}(U),$ every Radon measure becomes a distribution on {{mvar|U}}. If $f$ is a locally integrable function on {{mvar|U}} then the distribution $\backslash phi\; \backslash mapsto\; \backslash textstyle\; \backslash int\_U\; f(x)\; \backslash phi(x)\; \backslash ,\; dx$ is a Radon measure; so Radon measures form a large and important space of distributions.

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

{{math theorem|name=Theorem.{{sfn|Trèves|2006|pp=262–264}}|math_statement=

Suppose $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ is a Radon measure, where $U\; \backslash subseteq\; \backslash R^n,$ let $V\; \backslash subseteq\; U$ be a neighborhood of the support of $T,$ and let $I\; =\; \backslash \{p\; \backslash in\; \backslash N^n\; :\; |p|\; \backslash leq\; n\backslash \}.$ There exists a family $f=(f\_p)\_\{p\backslash in\; I\}$ of locally $L^\backslash infty$ functions on {{mvar|U}} such that $\backslash operatorname\{supp\}\; f\_p\; \backslash subseteq\; V$ for every $p\backslash in\; I,$ and

$$T\; =\; \backslash sum\_\{p\backslash in\; I\}\; \backslash partial^p\; f\_p.$$

Furthermore, $T$ is also equal to a finite sum of derivatives of continuous functions on $U,$ where each derivative has order $\backslash leq\; 2\; n.$

}}

Positive Radon measures

A linear function {{mvar|T}} on a space of functions is called {{em|positive}} if whenever a function $f$ that belongs to the domain of {{mvar|T}} is non-negative (meaning that $f$ is real-valued and $f\; \backslash geq\; 0$) then $T(f)\; \backslash geq\; 0.$ One may show that every positive linear functional on $C\_c^0(U)$ is necessarily continuous (that is, necessarily a Radon measure).{{sfn|Trèves|2006|p=218}}

Lebesgue measure is an example of a positive Radon measure.

One particularly important class of Radon measures are those that are induced locally integrable functions. The function $f\; :\; U\; \backslash to\; \backslash R$ is called {{em|locally integrable}} if it is Lebesgue integrable over every compact subset {{mvar|K}} of {{mvar|U}}.For more information on such class of functions, see the entry on locally integrable functions. This is a large class of functions which includes all continuous functions and all Lp space $L^p$ functions. The topology on $\backslash mathcal\{D\}(U)$ is defined in such a fashion that any locally integrable function $f$ yields a continuous linear functional on $\backslash mathcal\{D\}(U)$ – that is, an element of $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ – denoted here by $T\_f$, whose value on the test function $\backslash phi$ is given by the Lebesgue integral:

$$\backslash langle\; T\_f,\; \backslash phi\; \backslash rangle\; =\; \backslash int\_U\; f\; \backslash phi\backslash ,dx.$$

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

$$\backslash langle\; f,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T\_f,\; \backslash phi\; \backslash rangle.$$

If $f$ and {{mvar|g}} are two locally integrable functions, then the associated distributions $T\_f$ and {{mvar|T_g}} are equal to the same element of $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ if and only if $f$ and {{mvar|g}} are equal almost everywhere (see, for instance, {{harvtxt|Hörmander|1983|loc=Theorem 1.2.5}}). In a similar manner, every Radon measure $\backslash mu$ on {{mvar|U}} defines an element of $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ whose value on the test function $\backslash phi$ is $\backslash textstyle\backslash int\backslash phi\; \backslash ,d\backslash mu.$ As above, it is conventional to abuse notation and write the pairing between a Radon measure $\backslash mu$ and a test function $\backslash phi$ as $\backslash langle\; \backslash mu,\; \backslash phi\; \backslash rangle.$ Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

Test functions as distributions

The test functions are themselves locally integrable, and so define distributions. The space of test functions $C\_c^\backslash infty(U)$ is sequentially dense in $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ with respect to the strong topology on $\backslash mathcal\{D\}^\{\backslash prime\}(U).${{sfn|Trèves|2006|pp=300-304}} This means that for any $T\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(U),$ there is a sequence of test functions, $(\backslash phi\_i)\_\{i=1\}^\backslash infty,$ that converges to $T\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(U)$ (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,

$$\backslash langle\; \backslash phi\_i,\; \backslash psi\; \backslash rangle\; \backslash to\; \backslash langle\; T,\; \backslash psi\; \backslash rangle\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; \backslash psi\; \backslash in\; \backslash mathcal\{D\}(U).$$

Furthermore, $C\_c^\backslash infty(U)$ is also sequentially dense in the strong dual space of $C^\backslash infty(U).${{sfn|Trèves|2006|pp=300-304}}

The inclusion map $\backslash operatorname\{In\}\; :\; C\_c^\backslash infty(U)\; \backslash to\; C^\backslash infty(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $\{\}^\{t\}\backslash operatorname\{In\}\; :\; \backslash left(C^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b\; \backslash to\; \backslash mathcal\{D\}^\{\backslash prime\}(U)\; =\; \backslash left(C\_c^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b$ is also a continuous injection. Thus the image of the transpose, denoted by $\backslash mathcal\{E\}^\{\backslash prime\}(U),$ forms a space of distributions when it is endowed with the strong dual topology of $\backslash left(C^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b$ (transferred to it via the transpose map $\{\}^\{t\}\backslash operatorname\{In\}\; :\; \backslash left(C^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b\; \backslash to\; \backslash mathcal\{E\}^\{\backslash prime\}(U),$ so the topology of $\backslash mathcal\{E\}^\{\backslash prime\}(U)$ is finer than the subspace topology that this set inherits from $\backslash mathcal\{D\}^\{\backslash prime\}(U)$).{{sfn|Trèves|2006|pp=255-257}}

The elements of $\backslash mathcal\{E\}^\{\backslash prime\}(U)\; =\; \backslash left(C^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b$ can be identified as the space of distributions with compact support.{{sfn|Trèves|2006|pp=255-257}} Explicitly, if {{mvar|T}} is a distribution on {{mvar|U}} then the following are equivalent,

• $T\; \backslash in\; \backslash mathcal\{E\}^\{\backslash prime\}(U)$;
• the support of {{mvar|T}} is compact;
• the restriction of $T$ to $C\_c^\backslash infty(U),$ when that space is equipped with the subspace topology inherited from $C^\backslash infty(U)$ (a coarser topology than the canonical LF topology), is continuous;{{sfn|Trèves|2006|pp=255-257}}
• there is a compact subset {{mvar|K}} of {{mvar|U}} such that for every test function $\backslash phi$ whose support is completely outside of {{mvar|K}}, we have $T(\backslash phi)=0.$

Compactly supported distributions define continuous linear functionals on the space $C^\backslash infty(U)$; recall that the topology on $C^\backslash infty(U)$ is defined such that a sequence of test functions $\backslash phi\_k$ converges to 0 if and only if all derivatives of $\backslash phi\_k$ converge uniformly to 0 on every compact subset of {{mvar|U}}. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from $C\_c^\backslash infty(U)$ to $C^\backslash infty(U).$

Let $k\; \backslash in\; \backslash N.$ The inclusion map $\backslash operatorname\{In\}\; :\; C\_c^\backslash infty(U)\; \backslash to\; C\_c^k(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $\{\}^\{t\}\backslash operatorname\{In\}\; :\; \backslash left(C\_c^k(U)\backslash right)^\{\backslash prime\}\_b\; \backslash to\; \backslash mathcal\{D\}^\{\backslash prime\}(U)\; =\; \backslash left(C\_c^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b$ is also a continuous injection. Consequently, the image of $\{\}^\{t\}\backslash operatorname\{In\},$ denoted by $\backslash mathcal\{D\}\text{'}^k(U),$ forms a space of distributions when it is endowed with the strong dual topology of $\backslash left(C\_c^k(U)\backslash right)^\{\backslash prime\}\_b$ (transferred to it via the transpose map $\{\}^\{t\}\backslash operatorname\{In\}\; :\; \backslash left(C^\backslash infty(U)\backslash right)^\{\backslash prime\}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}^k(U),$ so $\backslash mathcal\{D\}\text{'}^\{m\}(U)$'s topology is finer than the subspace topology that this set inherits from $\backslash mathcal\{D\}^\{\backslash prime\}(U)$). The elements of $\backslash mathcal\{D\}\text{'}^k(U)$ are {{em|the distributions of order $\backslash ,\backslash leq\; k.$}}{{sfn|Trèves|2006|pp=258-264}} The distributions of order $\backslash ,\backslash leq\; 0,$ which are also called {{em|distributions of order $0,$}} are exactly the distributions that are Radon measures (described above).

For $0\; \backslash neq\; k\; \backslash in\; \backslash N,$ a {{em|distribution of order $k$}} is a distribution of order $\backslash ,\backslash leq\; k$ that is not a distribution of order $\backslash ,\backslash leq\; k\; -\; 1${{sfn|Trèves|2006|pp=258-264}}

A distribution is said to be of {{em|finite order}} if there is some integer {{mvar|k}} such that it is a distribution of order $\backslash ,\backslash leq\; k,$ and the set of distributions of finite order is denoted by $\backslash mathcal\{D\}\text{'}^\{F\}(U).$ Note that if $k\; \backslash leq\; 1$ then $\backslash mathcal\{D\}\text{'}^k(U)\; \backslash subseteq\; \backslash mathcal\{D\}\text{'}^\{l\}(U)$ so that $\backslash mathcal\{D\}\text{'}^\{F\}(U)$ is a vector subspace of $\backslash mathcal\{D\}^\{\backslash prime\}(U)$ and furthermore, if and only if $\backslash mathcal\{D\}\text{'}^\{F\}(U)\; =\; \backslash mathcal\{D\}^\{\backslash prime\}(U).${{sfn|Trèves|2006|pp=258-264}}

Structure of distributions of finite order

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

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

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=Suppose $T\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(U)$ has finite order and $I\; =\backslash \{p\; \backslash in\; \backslash N^n\; :\; |p|\; \backslash leq\; k\backslash \}.$ Given any open subset {{mvar|V}} of {{mvar|U}} containing the support of {{mvar|T}}, there is a family of Radon measures in {{mvar|U}}, $(\backslash mu\_p)\_\{p\; \backslash in\; I\},$ such that for very $p\; \backslash in\; I,\; \backslash operatorname\{supp\}(\backslash mu\_p)\; \backslash subseteq\; V$ and

$$T\; =\; \backslash sum\_\{|p|\; \backslash leq\; k\}\; \backslash partial^p\; \backslash mu\_p.$$}}

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

$$S\; f\; :=\; \backslash sum\_\{m=1\}^\backslash infty\; (\backslash partial^\{m\}\; f)\backslash left(\backslash frac\{1\}\{m\}\backslash right).$$

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

{{for|tempered distributions on semisimple groups|Tempered representation}}

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

Schwartz space

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

$$p\_\{\backslash alpha,\; \backslash beta\}\; (\backslash phi)\; ~=~\; \backslash sup\_\{x\; \backslash in\; \backslash R^n\}\; \backslash left|x^\backslash alpha\; \backslash partial^\backslash beta\; \backslash phi(x)\; \backslash right|.$$

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

The family of seminorms $p\_\{\backslash alpha,\backslash beta\}$ defines a locally convex topology on the Schwartz space. For $n\; =\; 1,$ the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:{{sfn|Trèves|2006|pp=92-94}}

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

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

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

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

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

$\backslash mathcal\{D\}(\backslash R^n)$ is dense in $\backslash mathcal\{S\}(\backslash R^n).$ The subset of all analytic Schwartz functions is dense in $\backslash mathcal\{S\}(\backslash R^n)$ as well.{{sfn|Trèves|2006|pp=160}}

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

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

where $\backslash widehat\{\backslash otimes\}$ represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product).{{sfn|Trèves|2006|p=531}}

Tempered distributions

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

The space $\backslash mathcal\{S\}^\{\backslash prime\}(\backslash R^n)$ is called the space of {{em|tempered distributions}}. It is the continuous dual of the Schwartz space. Equivalently, a distribution {{mvar|T}} is a tempered distribution if and only if

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

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

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

Fourier transform

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform $F\; :\; \backslash mathcal\{S\}(\backslash R^n)\; \backslash to\; \backslash mathcal\{S\}(\backslash R^n)$ is a TVS-automorphism of the Schwartz space, and the {{em|Fourier transform}} is defined to be its transpose $\{\}^\{t\}F\; :\; \backslash mathcal\{S\}^\{\backslash prime\}(\backslash R^n)\; \backslash to\; \backslash mathcal\{S\}^\{\backslash prime\}(\backslash R^n),$ which (abusing notation) will again be denoted by {{mvar|F}}. So the Fourier transform of the tempered distribution {{mvar|T}} is defined by $(FT)(\backslash psi)\; =\; T(F\; \backslash psi)$ for every Schwartz function $\backslash psi.$ $FT$ is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that

$$F\; \backslash dfrac\{dT\}\{dx\}\; =\; ixFT$$

and also with convolution: if {{mvar|T}} is a tempered distribution and $\backslash psi$ is a {{em|slowly increasing}} smooth function on $\backslash R^n,$ $\backslash psi\; T$ is again a tempered distribution and

$$F(\backslash psi\; T)\; =\; F\; \backslash psi\; *\; FT$$

is the convolution of $FT$ and $F\; \backslash psi$. In particular, the Fourier transform of the constant function equal to 1 is the $\backslash delta$ distribution.

Expressing tempered distributions as sums of derivatives

If $T\; \backslash in\; \backslash mathcal\{S\}^\{\backslash prime\}(\backslash R^n)$ is a tempered distribution, then there exists a constant $C\; >\; 0,$ and positive integers {{mvar|M}} and {{mvar|N}} such that for all Schwartz functions $\backslash phi\; \backslash in\; \backslash mathcal\{S\}(\backslash R^n)$

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

This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function {{mvar|F}} and a multi-index $\backslash alpha$ such that

$$T\; =\; \backslash partial^\backslash alpha\; F.$$

Restriction of distributions to compact sets

If $T\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(\backslash R^n),$ then for any compact set $K\; \backslash subseteq\; \backslash R^n,$ there exists a continuous function {{mvar|F}} compactly supported in $\backslash R^n$ (possibly on a larger set than {{mvar|K}} itself) and a multi-index $\backslash alpha$ such that $T\; =\; \backslash partial^\backslash alpha\; F$ on $C\_c^\backslash infty(K).$

Let $U\; \backslash subseteq\; \backslash R^m$ and $V\; \backslash subseteq\; \backslash R^n$ be open sets. Assume all vector spaces to be over the field $\backslash mathbb\{F\},$ where $\backslash mathbb\{F\}=\backslash R$ or $\backslash Complex.$ For $f\; \backslash in\; \backslash mathcal\{D\}(U\; \backslash times\; V)$ define for every $u\; \backslash in\; U$ and every $v\; \backslash in\; V$ the following functions:

$$\backslash begin\{alignat\}\{9\}$$

f_u : \,& V && \to \,&& \mathbb{F} && \quad \text{ and } \quad && f^v : \,&& U && \to \,&& \mathbb{F} \\

& y && \mapsto\,&& f(u, y) && && && x && \mapsto\,&& f(x, v) \\

\end{alignat}

Given $S\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(V),$ define the following functions:

$$\backslash begin\{alignat\}\{9\}$$

\langle S, f^{\bullet}\rangle : \,& V && \to \,&& \mathbb{F} && \quad \text{ and } \quad && \langle T, f_{\bullet}\rangle : \,&& U && \to \,&& \mathbb{F} \\

& v && \mapsto\,&& \langle S, f^v \rangle && && && u && \mapsto\,&& \langle T, f_u \rangle \\

\end{alignat}

where $\backslash langle\; T,\; f\_\{\backslash bullet\}\backslash rangle\; \backslash in\; \backslash mathcal\{D\}(U)$ and $\backslash langle\; S,\; f^\{\backslash bullet\}\backslash rangle\; \backslash in\; \backslash mathcal\{D\}(V).$

These definitions associate every $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(V)$ with the (respective) continuous linear map:

$$\backslash begin\{alignat\}\{9\}$$

\,& \mathcal{D}(U \times V) && \to \,&& \mathcal{D}(V) && \quad \text{ and } \quad && \,&& \mathcal{D}(U \times V) && \to \,&& \mathcal{D}(U) \\

& f && \mapsto\,&& \langle S, f^{\bullet} \rangle && && && f && \mapsto\,&& \langle T, f_{\bullet} \rangle \\

\end{alignat}

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

{{Math theorem|name=Fubini's theorem for distributions{{sfn|Trèves|2006|pp=416-419}}|math_statement=

Let $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(V).$ If $f\; \backslash in\; \backslash mathcal\{D\}(U\; \backslash times\; V)$ then

$$\backslash langle\; S,\; \backslash langle\; T,\; f\_\{\backslash bullet\}\; \backslash rangle\; \backslash rangle\; =\; \backslash langle\; T,\; \backslash langle\; S,\; f^\{\backslash bullet\}\; \backslash rangle\; \backslash rangle.$$

}}

{{em|The tensor product of $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(V),$}} denoted by $S\; \backslash otimes\; T$ or $T\; \backslash otimes\; S,$ is the distribution in $U\; \backslash times\; V$ defined by:{{sfn|Trèves|2006|pp=416-419}}

$$(S\; \backslash otimes\; T)(f)\; :=\; \backslash langle\; S,\; \backslash langle\; T,\; f\_\{\backslash bullet\}\; \backslash rangle\; \backslash rangle\; =\; \backslash langle\; T,\; \backslash langle\; S,\; f^\{\backslash bullet\}\backslash rangle\; \backslash rangle.$$

The tensor product defines a bilinear map

$$\backslash begin\{alignat\}\{4\}$$

\,& \mathcal{D}^{\prime}(U) \times \mathcal{D}^{\prime}(V) && \to \,&& \mathcal{D}^{\prime}(U \times V) \\

& ~~~~~~~~(S, T) && \mapsto\,&& S \otimes T \\

\end{alignat}

the span of the range of this map is a dense subspace of its codomain. Furthermore, $\backslash operatorname\{supp\}\; (S\; \backslash otimes\; T)\; =\; \backslash operatorname\{supp\}(S)\; \backslash times\; \backslash operatorname\{supp\}(T).${{sfn|Trèves|2006|pp=416-419}} Moreover $(S,T)\; \backslash mapsto\; S\; \backslash otimes\; T$ induces continuous bilinear maps:

$$\backslash begin\{alignat\}\{8\}$$

&\mathcal{E}^{\prime}(U) &&\times \mathcal{E}^{\prime}(V) &&\to \mathcal{E}^{\prime}(U \times V) \\

&\mathcal{S}^{\prime}(\R^m) &&\times \mathcal{S}^{\prime}(\R^n) &&\to \mathcal{S}^{\prime}(\R^{m + n}) \\

\end{alignat}

where $\backslash mathcal\{E\}\text{'}$ denotes the space of distributions with compact support and $\backslash mathcal\{S\}$ is the Schwartz space of rapidly decreasing functions.{{sfn|Trèves|2006|p=423}}

{{Math theorem|name=Schwartz kernel theorem{{sfn|Trèves|2006|p=531}}|math_statement=

Each of the canonical maps below (defined in the natural way) are TVS isomorphisms:

$$\backslash begin\{alignat\}\{8\}$$

&\mathcal{S}^{\prime}(\R^{m+n}) ~&&~\cong~&&~ \mathcal{S}^{\prime}(\R^m) \ &&\widehat{\otimes}\ \mathcal{S}^{\prime}(\R^n) ~&&~\cong~&&~ L_b(\mathcal{S}(\R^m); &&\;\mathcal{S}^{\prime}(\R^n)) \\

&\mathcal{E}^{\prime}(U \times V) ~&&~\cong~&&~ \mathcal{E}^{\prime}(U) \ &&\widehat{\otimes}\ \mathcal{E}^{\prime}(V) ~&&~\cong~&&~ L_b(C^\infty(U); &&\;\mathcal{E}^{\prime}(V)) \\

&\mathcal{D}^{\prime}(U \times V) ~&&~\cong~&&~ \mathcal{D}^{\prime}(U) \ &&\widehat{\otimes}\ \mathcal{D}^{\prime}(V) ~&&~\cong~&&~ L_b(\mathcal{D}(U); &&\;\mathcal{D}^{\prime}(V)) \\

\end{alignat}

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

}}

This result does not hold for Hilbert spaces such as $L^2$ and its dual space.{{sfn|Trèves|2006|pp=509-510}} Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space $L^2$? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because $\backslash mathcal\{D\}(U)$ is a nuclear space that the Schwartz kernel theorem holds. Like Hilbert spaces, nuclear spaces may be thought as of generalizations of finite dimensional Euclidean space.

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

• {{annotated link|Colombeau algebra}}
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• {{annotated link|Distribution on a linear algebraic group}}
• {{annotated link|Gelfand triple}}
• {{annotated link|Generalized function}}
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• {{annotated link|Laplacian of the indicator}}
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• {{annotated link|Malgrange–Ehrenpreis theorem}}
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{{reflist|group=note}}

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{{Functional analysis}}

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Category:Functional analysis

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