The “fundamental operations” on distributions (pullback of distributions, product of distributions) are in general not continuous in the subspace topology, for subspaces where Hörmander's criterion is satisfied, of the canonical dual topological vector space-topology (“weak topology”).
There is however a pseudo-topology which respect to which the fundamental operations are continuous. This is due to Hörmander 71, p. 125 and hence called the Hörmander topology.
Original articles include
Lars Hörmander, Fourier integral operators. I , Acta Math. 127 (1971), 79–183
Yoann Dabrowski, Christian Brouder, Functional properties of Hörmander’s space of distributions having a specified wavefront set (arXiv:1308.1061)
Christian Brouder, Nguyen Viet Dang, Frédéric Hélein, Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker), Studia Mathematica 232 (2016), 201-226 (arXiv:org)
Review in the context of causal perturbation theory/perturbative QFT is in
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