= Continuity of distributional differentiation
{title2=$u_j\to u\text{ in }\mathcal D'\Rightarrow D^\alpha u_j\to D^\alpha u$}
If $u_j\to u$ in <weak convergence of distributions>, then every fixed <distributional derivative> converges too. For each <test function> $\phi$, its derivative $D^\alpha\phi$ is another <test function>, and
$$
\langle D^\alpha u_j,\phi\rangle=(-1)^{|\alpha|}\langle u_j,D^\alpha\phi\rangle
\longrightarrow(-1)^{|\alpha|}\langle u,D^\alpha\phi\rangle.
$$
Thus a series converging as <distributions> can be differentiated term by term in that topology. For example, <uniform convergence> on <compact sets> implies convergence of the associated <regular distributions>, even when the ordinary series of <derivatives> does not converge pointwise.
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Target: Even-weight principal-value distribution. Parent: Principal-value reciprocal distribution. Intended file: distribution-theory.bigb.
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