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$ means that the topological vector space $ A$ is continuously embedded in the topological vector space $ B$ . This is typically valid for the norm $ \lVert \cdot \rVert_{L^2}$ but not for the norm $ \lVert \mathrm{D} \cdot \rVert_{L^2}$ , where $ \mathrm{D}$ is the derivative operator.

NB. Here, the space of tempered generalized functions $ \mathcal{S}'(\mathbb{R})$ is endowed with the weak topology.