# Wk* continuity and wk* sequential continuity on $S'$

Let $$S$$ be the Schwartz space and $$S'$$ be its dual space. If a linear operator $$T:S'\rightarrow S'$$ is weakly* sequentially continuous, is it also weakly* continuous?

Yes. Since boundedness is determined by sequences, $$T$$ maps bounded (= equicontinuous) sets of $$S'$$ to bounded sets. Since the Schwartz space $$S$$ is a Frechet-Schwartz space (i.e., a projective limit of Banach spaces with compact linking maps), either by results of Grothendieck or a theorem of Laurent Schwartz, $$S'$$ endowed with the strong topology $$\beta(S',S)$$ (of uniform convergence on bounded subsets of $$S$$) is bornological and hence $$T$$ is continuous as a map on $$(S',\beta(S',S))$$. Then, the transposed $$T^t$$ is a continuous map on $$S''=S$$ (Frechet-Schwartz spaces are reflexive) and thus, $$T=T^{tt}$$ is weak$$^*$$-continuous on $$S$$.