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Assume that $ \lVert \cdot \rVert$ is a norm on the space of rapidly decaying functions $ \mathcal{S}(\mathbb{R})$ . Under which conditions on the norm can we say that the completion $ \mathcal{X}$ for this norm is a Banach space such that \begin{equation} \mathcal{S}(\mathbb{R}) \subseteq \mathcal{X} \subseteq \mathcal{S}'(\mathbb{R}), \end{equation} where $ A \subseteq B$ meansRead more