Continuity of distributional differentiation ($u_j\to u\text{ in }{\mathcal{D}}^{\prime }\Rightarrow D^{\alpha }{u}_j\to D^{\alpha }u$)

If $u_j\to u$ in weak convergence of distributions, then every fixed distributional derivative converges too. For each test function $\varphi$, its derivative $D^{\alpha }\varphi$ is another test function, andThus 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.

Target: Even-weight principal-value distribution. Parent: Principal-value reciprocal distribution. Intended file: distribution-theory.bigb.