Proposition: Descent

Let $P:{\mathcal{D}}^{\prime }(X)\to {\mathcal{D}}^{\prime }(Y)$ be continuous and linear, and let $\mathfrak{C}\subseteq {\mathcal{D}}^{\prime }(X)$ and ${\mathfrak{C}}_P\subseteq {\mathcal{D}}^{\prime }(Y)$ be declared real linear subspaces. The assignment

is well defined on the real vector-space quotients if and only if

With quotient topologies, the induced map is continuous.

Proof. If $u-v\in \mathfrak{C}$, then $Pu-Pv=P(u-v)$. Condition eq:descent therefore makes the output class independent of the representative. Conversely, well-definedness applied to $[c]=[0]$, $c\in \mathfrak{C}$, gives $Pc\in {\mathfrak{C}}_P$. Continuity follows from the defining property of the quotient topology.