Editing Distribution (mathematics) (section)

===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>