Editing Distribution (mathematics) (section)

===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]]).