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Let $\mathcal{E}'(\mathbb{R})$ be equipped with its usual strong topology (being the dual space of $\mathcal{E}(\mathbb{R})$). Is convolution jointly continuous on $\mathcal{E}'(\mathbb{R})$?

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Yes. This is Théorème IV in §3 of Chapitre VI (page 157) in Laurent Schwartz's Théorie des distributions.

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