Editing Distribution (mathematics) (section)

===Tensor products of distributions{{anchor|Tensor product of distributions}}===

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

Given <math>S \in \mathcal{D}^{\prime}(U)</math> and <math>T \in \mathcal{D}^{\prime}(V),</math> define the following functions:
<math display=block>\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}</math>
where <math>\langle T, f_{\bullet}\rangle \in \mathcal{D}(U)</math> and <math>\langle S, f^{\bullet}\rangle \in \mathcal{D}(V).</math> 
These definitions associate every <math>S \in \mathcal{D}'(U)</math> and <math>T \in \mathcal{D}'(V)</math> with the (respective) continuous linear map:
<math display=block>\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}</math>

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

{{Math theorem|name={{visible anchor|Fubini's theorem for distributions|text=[[Fubini's theorem]] for distributions}}{{sfn|Trèves|2006|pp=416-419}}|math_statement=
Let <math>S \in \mathcal{D}'(U)</math> and <math>T \in \mathcal{D}'(V).</math> If <math>f \in \mathcal{D}(U \times V)</math> then 
<math display=block>\langle S, \langle T, f_{\bullet} \rangle \rangle = \langle T, \langle S, f^{\bullet} \rangle \rangle.</math>
}}

{{em|The [[Tensor product|'''{{visible anchor|tensor product}}''']] of <math>S \in \mathcal{D}'(U)</math> and <math>T \in \mathcal{D}'(V),</math>}} denoted by <math>S \otimes T</math> or <math>T \otimes S,</math> is the distribution in <math>U \times V</math> defined by:{{sfn|Trèves|2006|pp=416-419}}
<math display=block>(S \otimes T)(f) := \langle S, \langle T, f_{\bullet} \rangle \rangle = \langle T, \langle S, f^{\bullet}\rangle \rangle.</math>