Editing Distribution (mathematics) (section)

==Definitions of test functions and distributions==

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

{{block indent|em=1.5|text='''Notation''': 
# Let <math>k \in \{0, 1, 2, \ldots, \infty\}.</math>
# Let <math>C^k(U)</math> denote the [[vector space]] of all {{mvar|k}}-times [[continuously differentiable]] real or complex-valued functions on {{mvar|U}}.
# For any compact subset <math>K \subseteq U,</math> let <math>C^k(K)</math> and <math>C^k(K;U)</math> both denote the vector space of all those functions <math>f \in C^k(U)</math> such that <math>\operatorname{supp}(f) \subseteq K.</math>
#* If <math>f \in C^k(K)</math> then the domain of <math>f</math> is {{mvar|U}} and not {{mvar|K}}. So although <math>C^k(K)</math> depends on both {{mvar|K}} and {{mvar|U}}, only {{mvar|K}} is typically indicated. The justification for this common practice is [[#Omitting the open set from notation|detailed below]]. The notation <math>C^k(K;U)</math> will only be used when the notation <math>C^k(K)</math> risks being ambiguous.
#* Every <math>C^k(K)</math> contains the constant {{math|0}} map, even if <math>K = \varnothing.</math>
# Let <math>C_c^k(U)</math> denote the set of all <math>f \in C^k(U)</math> such that <math>f \in C^k(K)</math> for some compact subset {{mvar|K}} of {{mvar|U}}.
#* Equivalently, <math>C_c^k(U)</math> is the set of all <math>f \in C^k(U)</math> such that <math>f</math> has compact [[#support of a function|support]].
#* <math>C_c^k(U)</math> is equal to the union of all <math>C^k(K)</math> as <math>K \subseteq U</math> ranges over all compact subsets of <math>U.</math>
#* If <math>f</math> is a real-valued function on <math>U</math>, then <math>f</math> is an element of <math>C_c^k(U)</math> if and only if <math>f</math> is a <math>C^k</math> [[bump function]]. Every real-valued test function on <math>U</math> is also a complex-valued test function on <math>U.</math>
}}

[[File:Bump.png|thumb|350x350px|The graph of the [[bump function]] <math>(x,y) \in \R^2 \mapsto \Psi(r),</math> where <math>r = \left(x^2 + y^2\right)^\frac{1}{2}</math> and <math>\Psi(r) = e^{-\frac{1}{1 - r^2}}\cdot\mathbf{1}_{\{|r|<1\}}.</math> This function is a test function on <math>\R^2</math> and is an element of <math>C^\infty_c\left(\R^2\right).</math> The [[#support of a function|support]] of this function is the closed [[unit disk]] in <math>\R^2.</math> It is non-zero on the open unit disk and it is equal to {{math|0}} everywhere outside of it.]]

For all <math>j, k \in \{0, 1, 2, \ldots, \infty\}</math> and any compact subsets <math>K</math> and <math>L</math> of <math>U</math>, we have:
<math display=block>\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 \le k \\
C_c^k(U) &\subseteq C^j_c(U) && \text{if } j \le k \\
C^k(U) &\subseteq C^j(U) && \text{if } j \le k \\
\end{align}</math>

{{block indent|em=1.5|text='''Definition''': Elements of <math>C_c^\infty(U)</math> are called '''{{em|test functions}}''' on {{mvar|U}} and <math>C_c^\infty(U)</math> is called the '''{{em|space of test functions}}''' on {{mvar|U}}. We will use both <math>\mathcal{D}(U)</math> and <math>C_c^\infty(U)</math> to denote this space.}}

Distributions on {{mvar|U}} are [[continuous linear functional]]s on <math>C_c^\infty(U)</math> when this vector space is endowed with a particular topology called the '''{{em|canonical LF-topology}}'''. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on <math>C_c^\infty(U)</math> that are often straightforward to verify.

'''Proposition''': A [[Linear form|linear functional]] {{mvar|T}} on <math>C_c^\infty(U)</math> is continuous, and therefore a '''{{em|distribution}}''', if and only if any of the following equivalent conditions is satisfied:

# For every compact subset <math>K\subseteq U</math> there exist constants <math>C>0</math> and <math>N\in \N</math> (dependent on <math>K</math>) such that for all <math>f \in C_c^\infty(U)</math> with [[#support of a function|support]] contained in <math>K</math>,{{sfn|Trèves|2006|pp=222-223}}<ref>{{harvnb|Grubb|2009|page=14}}</ref> <math display="block">|T(f)| \leq C \sup \{|\partial^\alpha f(x)|: x \in U, |\alpha| \leq N\}.</math>
# For every compact subset <math>K\subseteq U</math> and every sequence <math>\{f_i\}_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> whose supports are contained in <math>K</math>, if <math>\{\partial^\alpha f_i\}_{i=1}^\infty</math> converges uniformly to zero on <math>U</math> for every [[multi-index]] <math>\alpha</math>, then <math>T(f_i) \to 0.</math>

===Topology on ''C''<sup>''k''</sup>(''U'')===

We now introduce the [[seminorm]]s that will define the topology on <math>C^k(U).</math> 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 <math>k \in \{0, 1, 2, \ldots, \infty\}</math> and <math>K</math> is an arbitrary compact subset of <math>U.</math> Suppose <math>i</math> is an integer such that <math>0 \leq i \leq k</math><ref group=note>Note that <math>i</math> being an integer implies <math>i \neq \infty.</math> This is sometimes expressed as <math>0 \leq i < k + 1.</math> Since <math>\infty + 1 = \infty,</math> the inequality "<math>0 \leq i < k + 1</math>" means: <math>0 \leq i < \infty</math> if <math>k = \infty,</math> while if <math>k \neq \infty</math> then it means <math>0 \leq i \leq k.</math></ref> and <math>p</math> is a multi-index with length <math>| p|\leq k.</math> For <math>K \neq \varnothing</math> and <math>f \in C^k(U),</math> define:
<math display=block>\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}</math>
while for <math>K = \varnothing,</math> define all the functions above to be the constant {{math|0}} map. 
}}
All of the functions above are non-negative <math>\R</math>-valued<ref group="note">The image of the [[compact set]] <math>K</math> under a continuous <math>\R</math>-valued map (for example, <math>x \mapsto \left|\partial^p f(x)\right|</math> for <math>x \in U</math>) is itself a compact, and thus bounded, subset of <math>\R.</math> If <math>K \neq \varnothing</math> then this implies that each of the functions defined above is <math>\R</math>-valued (that is, none of the [[Infimum and supremum|supremums]] above are ever equal to <math>\infty</math>).</ref> [[seminorm]]s on <math>C^k(U).</math> As explained in [[Locally convex topological vector space#Definition via seminorms|this article]], every set of seminorms on a vector space induces a [[Locally convex topological vector space|locally convex]] [[Topological vector space|vector topology]].

Each of the following sets of seminorms 
<math display=block>\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}</math>
generate the same [[Locally convex topological vector space|locally convex]] [[Topological vector space|vector topology]] on <math>C^k(U)</math> (so for example, the topology generated by the seminorms in <math>A</math> is equal to the topology generated by those in <math>C</math>). 

{{block indent|em=1.5|text=The vector space <math>C^k(U)</math> is endowed with the [[Locally convex topological vector space|locally convex]] topology induced by any one of the four families <math>A, B, C, D</math> of seminorms described above. This topology is also equal to the vector topology induced by {{em|all}} of the seminorms in <math>A \cup B \cup C \cup D.</math>}}

With this topology, <math>C^k(U)</math> becomes a locally convex [[Fréchet space]] that is {{em|not}} [[Normable space|normable]]. Every element of <math>A \cup B \cup C \cup D</math> is a continuous seminorm on <math>C^k(U).</math>
Under this topology, a [[net (mathematics)|net]] <math>(f_i)_{i\in I}</math> in <math>C^k(U)</math> converges to <math>f \in C^k(U)</math> if and only if for every multi-index <math>p</math> with <math>|p|< k + 1</math> and every compact <math>K,</math> the net of partial derivatives <math>\left(\partial^p f_i\right)_{i \in I}</math> [[Uniform convergence|converges uniformly]] to <math>\partial^p f</math> on <math>K.</math>{{sfn|Trèves|2006|pp=85-89}} For any <math>k \in \{0, 1, 2, \ldots, \infty\},</math> any [[Bounded set (topological vector space)|(von Neumann) bounded subset]] of <math>C^{k+1}(U)</math> is a [[relatively compact]] subset of <math>C^k(U).</math>{{sfn|Trèves|2006|pp=142-149}} In particular, a subset of <math>C^\infty(U)</math> is bounded if and only if it is bounded in <math>C^i(U)</math> for all <math>i \in \N.</math>{{sfn|Trèves|2006| pp=142-149}} The space <math>C^k(U)</math> is a [[Montel space]] if and only if <math>k = \infty.</math>{{sfn|Trèves|2006|pp=356-358}}

A subset <math>W</math> of <math>C^\infty(U)</math> is open in this topology if and only if there exists <math>i\in \N</math> such that <math>W</math> is open when <math>C^\infty(U)</math> is endowed with the [[subspace topology]] induced on it by <math>C^i(U).</math>

====Topology on ''C''<sup>''k''</sup>(''K'')====

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

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

If <math>k</math> is finite then <math>C^k(K)</math> is a [[Banach space]]{{sfn|Trèves|2006|pp=131-134}} with a topology that can be defined by the [[Norm (mathematics)|norm]]
<math display=block>r_K(f) := \sup_{|p|<k} \left( \sup_{x_0 \in K} \left|\partial^p f(x_0)\right| \right).</math>

====Trivial extensions and independence of ''C''<sup>''k''</sup>(''K'')'s topology from ''U''====
{{anchor|Omitting the open set from notation}}

Suppose <math>U</math> is an open subset of <math>\R^n</math> and <math>K \subseteq U</math> is a compact subset. By definition, elements of <math>C^k(K)</math> are functions with domain <math>U</math> (in symbols, <math>C^k(K) \subseteq C^k(U)</math>), so the space <math>C^k(K)</math> and its topology depend on <math>U;</math> to make this dependence on the open set <math>U</math> clear, temporarily denote <math>C^k(K)</math> by <math>C^k(K;U).</math> 
Importantly, changing the set <math>U</math> to a different open subset <math>U'</math> (with <math>K \subseteq U'</math>) will change the set <math>C^k(K)</math> from <math>C^k(K;U)</math> to <math>C^k(K;U'),</math><ref group="note">Exactly as with <math>C^k(K;U),</math> the space <math>C^k(K; U')</math> is defined to be the vector subspace of <math>C^k(U')</math> consisting of maps with [[#support of a function|support]] contained in <math>K</math> endowed with the subspace topology it inherits from <math>C^k(U')</math>.</ref> so that elements of <math>C^k(K)</math> will be functions with domain <math>U'</math> instead of <math>U.</math> 
Despite <math>C^k(K)</math> depending on the open set (<math>U \text{ or } U'</math>), the standard notation for <math>C^k(K)</math> makes no mention of it. 
This is justified because, as this subsection will now explain, the space <math>C^k(K;U)</math> is canonically identified as a subspace of <math>C^k(K;U')</math> (both algebraically and topologically). 

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

Given <math>f \in C_c^k(U),</math> its {{em|'''trivial extension''' to <math>V</math>}} is the function <math>F : V \to \Complex</math> defined by:
<math display=block>F(x) = \begin{cases}
f(x) & x \in U, \\
0 & \text{otherwise}.
\end{cases}</math>
This trivial extension belongs to <math>C^k(V)</math> (because <math>f \in C_c^k(U)</math> has compact support) and it will be denoted by <math>I(f)</math> (that is, <math>I(f) := F</math>). The assignment <math>f \mapsto I(f)</math> thus induces a map <math>I : C_c^k(U) \to C^k(V)</math> that sends a function in <math>C_c^k(U)</math> to its trivial extension on <math>V.</math> This map is a linear [[Injective function|injection]] and for every compact subset <math>K \subseteq U</math> (where <math>K</math> is also a compact subset of <math>V</math> since <math>K \subseteq U \subseteq V</math>), 
<math display=block>\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}</math>
If <math>I</math> is restricted to <math>C^k(K; U)</math> then the following induced linear map is a [[homeomorphism]] (linear homeomorphisms are called {{em|[[TVS-isomorphism]]s}}):
<math display=block>\begin{alignat}{4}
 \,& C^k(K; U) && \to \,&& C^k(K;V) \\
   & f                  && \mapsto\,&& I(f) \\
\end{alignat}</math>
and thus the next map is a [[topological embedding]]:
<math display=block>\begin{alignat}{4}
 \,& C^k(K; U) && \to \,&& C^k(V) \\
   & f                  && \mapsto\,&& I(f). \\
\end{alignat}</math>
Using the injection
<math display=block>I : C_c^k(U) \to C^k(V)</math>
the vector space <math>C_c^k(U)</math> is canonically identified with its image in <math>C_c^k(V) \subseteq C^k(V).</math> Because <math>C^k(K; U) \subseteq C_c^k(U),</math> through this identification, <math>C^k(K; U)</math> can also be considered as a subset of <math>C^k(V).</math>
Thus the topology on <math>C^k(K;U)</math> is independent of the open subset <math>U</math> of <math>\R^n</math> that contains <math>K,</math>{{sfn|Rudin|1991|pp=149-181}} which justifies the practice of writing <math>C^k(K)</math> instead of <math>C^k(K; U).</math>

===Canonical LF topology===
{{Main|Spaces of test functions and distributions}}
{{See also|LF-space|Topology of uniform convergence}}

Recall that <math>C_c^k(U)</math> denotes all functions in <math>C^k(U)</math> that have compact [[#support of a function|support]] in <math>U,</math> where note that <math>C_c^k(U)</math> is the union of all <math>C^k(K)</math> as <math>K</math> ranges over all compact subsets of <math>U.</math> Moreover, for each <math>k,\, C_c^k(U)</math> is a dense subset of <math>C^k(U).</math> The special case when <math>k = \infty</math> gives us the space of test functions.

{{block indent|em=1.5|text=<math>C_c^\infty(U)</math> is called the {{em|'''space of test functions''' on <math>U</math>}} and it may also be denoted by <math>\mathcal{D}(U).</math> Unless indicated otherwise, it is endowed with a topology called '''{{em|the canonical LF topology}}''', whose definition is given in the article: [[Spaces of test functions and distributions]].}}

The canonical LF-topology is {{em|not}} metrizable and importantly, it is [[Comparison of topologies|{{em|'''strictly''' finer}}]] than the [[subspace topology]] that <math>C^\infty(U)</math> induces on <math>C_c^\infty(U).</math> However, the canonical LF-topology does make <math>C_c^\infty(U)</math> into a [[Complete topological vector space|complete]] [[Reflexive space|reflexive]] [[Nuclear space|nuclear]]{{sfn|Trèves|2006|pp=526-534}} [[Montel space|Montel]]{{sfn|Trèves|2006|p=357}} [[Bornological space|bornological]] [[Barrelled space|barrelled]] [[Mackey space]]; the same is true of its [[strong dual space]] (that is, the space of all distributions with its usual topology). The canonical [[LF-space|LF-topology]] can be defined in various ways.

===Distributions===

{{See also|Continuous linear functional}}

As discussed earlier, continuous [[Linear form|linear functionals]] on a <math>C_c^\infty(U)</math> are known as distributions on <math>U.</math> Other equivalent definitions are described below.

{{block indent|em=1.5|text=By definition, a {{em|'''distribution''' on <math>U</math>}} is a [[Continuous linear operator|continuous]] [[Linear form|linear functional]] on <math>C_c^\infty(U).</math> Said differently, a distribution on <math>U</math> is an element of the [[continuous dual space]] of <math>C_c^\infty(U)</math> when <math>C_c^\infty(U)</math> is endowed with its canonical LF topology.}}

There is a canonical [[Dual system|duality pairing]] between a distribution <math>T</math> on <math>U</math> and a test function <math>f \in C_c^\infty(U),</math> which is denoted using [[angle brackets]] by
<math display=block>\begin{cases}
\mathcal{D}'(U) \times C_c^\infty(U) \to \R \\
(T, f) \mapsto \langle T, f \rangle := T(f)
\end{cases}</math>

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

====Characterizations of distributions====

'''Proposition.''' If <math>T</math> is a [[Linear form|linear functional]] on <math>C_c^\infty(U)</math> then the following are equivalent:

# {{mvar|T}} is a distribution;
# {{mvar|T}} is [[Continuous function|continuous]];
# {{mvar|T}} is [[Continuous function|continuous]] at the origin;
# {{mvar|T}} is [[Uniform continuity|uniformly continuous]];
# {{mvar|T}} is a [[bounded operator]];
# {{mvar|T}} is [[sequentially continuous]];
#* explicitly, for every sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> that converges in <math>C_c^\infty(U)</math> to some <math>f \in C_c^\infty(U),</math> <math display=inline>\lim_{i \to \infty} T\left(f_i\right) = T(f);</math><ref group="note">Even though the topology of <math>C_c^\infty(U)</math> is not metrizable, a linear functional on <math>C_c^\infty(U)</math> is continuous if and only if it is sequentially continuous.</ref>
# {{mvar|T}} is [[sequentially continuous]] at the origin; in other words, {{mvar|T}} maps null sequences<ref group=note name="Def null sequence">A '''{{em|null sequence}}''' is a sequence that converges to the origin.</ref> to null sequences;
#* explicitly, for every sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> that converges in <math>C_c^\infty(U)</math> to the origin (such a sequence is called a {{em|null sequence}}), <math display=inline>\lim_{i \to \infty} T\left(f_i\right) = 0;</math>
#* a {{em|null sequence}} is by definition any sequence that converges to the origin;
# {{mvar|T}} maps null sequences to bounded subsets;
#* explicitly, for every sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U)</math> that converges in <math>C_c^\infty(U)</math> to the origin, the sequence <math>\left(T\left(f_i\right)\right)_{i=1}^\infty</math> is bounded;
# {{mvar|T}} maps [[Mackey convergence|Mackey convergent]] null sequences to bounded subsets;
#* explicitly, for every Mackey convergent null sequence <math>\left(f_i\right)_{i=1}^\infty</math> in <math>C_c^\infty(U),</math> the sequence <math>\left(T\left(f_i\right)\right)_{i=1}^\infty</math> is bounded;
#* a sequence <math>f_{\bull} = \left(f_i\right)_{i=1}^\infty</math> is said to be {{em|[[Mackey convergence|Mackey convergent]] to the origin}} if there exists a divergent sequence <math>r_{\bull} = \left(r_i\right)_{i=1}^\infty \to \infty</math> of positive real numbers such that the sequence <math>\left(r_i f_i\right)_{i=1}^\infty</math> is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
# The kernel of {{mvar|T}} is a closed subspace of <math>C_c^\infty(U);</math>
# The graph of {{mvar|T}} is closed;
<!------ START: Removed information ----
* Note in particular that the following seminorms on <math>C^\infty(U)</math> (which recall were defined earlier) all restrict to continuous seminorms on <math>C_c^\infty(U)</math>: <math>\ q_{i,K}, \ r_{i,K}, \ r_{t,K},</math> and <math>s_{p,K},</math> where {{mvar|K}} is any compact subset of {{mvar|U}}, <math>i \geq 0</math> is an integer, and <math>p</math> is a multi-index. So to show that {{mvar|T}} is continuous, it {{em|suffices}} to show that the restriction to <math>C_c^\infty(U)</math> of one of these seminorms, call it <math>g,</math> satisfies <math>|T| \leq C g</math> for some <math>C > 0.</math>
---- END: Removed information ---->
# There exists a continuous seminorm <math>g</math> on <math>C_c^\infty(U)</math> such that <math>|T| \leq g;</math>
# There exists a constant <math>C > 0</math> and a finite subset <math>\{g_1, \ldots, g_m\} \subseteq \mathcal{P}</math> (where <math>\mathcal{P}</math> is any collection of continuous seminorms that defines the canonical LF topology on <math>C_c^\infty(U)</math>) such that <math>|T| \leq C(g_1 + \cdots + g_m);</math><ref group="note">If <math>\mathcal{P}</math> is also [[Directed set|directed]] under the usual function comparison then we can take the finite collection to consist of a single element.</ref>
# For every compact subset <math>K\subseteq U</math> there exist constants <math>C>0</math> and <math>N\in \N</math> such that for all <math>f \in C^\infty(K),</math>{{sfn|Trèves|2006|pp=222-223}} <math display=block>|T(f)| \leq C \sup \{|\partial^\alpha f(x)| : x \in U, |\alpha|\leq N\};</math>
# For every compact subset <math>K\subseteq U</math> there exist constants <math>C_K>0</math> and <math>N_K\in \N</math> such that for all <math>f \in C_c^\infty(U)</math> with [[#support of a function|support]] contained in <math>K,</math><ref name="Grubb 2009 page=14">See for example {{harvnb|Grubb|2009|page=14}}.</ref> <math display=block>|T(f)| \leq C_K \sup \{|\partial^\alpha f(x)| : x \in K, |\alpha|\leq N_K\};</math>
# For any compact subset <math>K\subseteq U</math> and any sequence <math>\{f_i\}_{i=1}^\infty</math> in <math>C^\infty(K),</math> if <math>\{\partial^p f_i\}_{i=1}^\infty</math> converges uniformly to zero for all [[multi-index|multi-indices]] <math>p,</math> then <math>T(f_i) \to 0;</math>

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

The set of all distributions on <math>U</math> is the [[continuous dual space]] of <math>C_c^\infty(U),</math> which when endowed with the [[Strong topology (polar topology)|strong dual topology]] is denoted by <math>\mathcal{D}'(U).</math> Importantly, unless indicated otherwise, the topology on <math>\mathcal{D}'(U)</math> is the [[strong dual topology]]; if the topology is instead the [[weak-* topology]] then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes <math>\mathcal{D}'(U)</math> into a [[Complete topological vector space|complete]] [[nuclear space]], to name just a few of its desirable properties.

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

A [[Linear map|{{em|linear}} map]] from <math>\mathcal{D}'(U)</math> into another [[locally convex topological vector space]] (such as any [[normed space]]) is [[Continuous function (topology)|continuous]] if and only if it is [[sequentially continuous]] at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general [[topological space]]s (for example, that are not also locally convex [[topological vector space]]s). The same is true of maps from <math>C_c^\infty(U)</math> (more generally, this is true of maps from any locally convex [[bornological space]]).