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The space $\mathcal{S}$ of rapidly decaying functions is endowed with its usual Fréchet topology. Its topological dual $\mathcal{S}'$ is endowed with the strong topology.

I assume I have a norm $\lVert \cdot \rVert$ on $\mathcal{S}$ that is continuous over $\mathcal{S}$ and denote by $\mathcal{X}$ the Banach space obtained as the completion of $(\mathcal{S},\lVert \cdot \rVert)$.

Is it correct to say that we have the embeddings $$\mathcal{S}\subseteq \mathcal{X} \subseteq \mathcal{S}'?$$ What if $\lVert \cdot \rVert$ is only a semi-norm?

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No, this isn't correct. To make things easier consider the "discrete" case of the rapidly decreasing sequences $s$ so that $s'$ is the space of slowly increasing sequences. As a norm on $s$ take $\|x\|= \sum_{n=1}^\infty |x_n|/e^n$. The completion of $(s,\|\cdot\|)$ is then the weighted $\ell^1$-space of all sequences with $\sum_{n=1}^\infty |x_n|/e^n$ finite which is not contained in $s'$.

The point is, that the inclusion $(s,\|\cdot\|) \hookrightarrow s'$ is not continuous. If $\|\cdot\|$ is only a seminorm, $\cal X$ should be the Hausdorff completion of $(s,\|\cdot\|)$ and the canonocal map $s\to \cal X$ will no longer be injective.

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  • $\begingroup$ Nice simple example. Thank you. If we write $s' = \cup s_n$ with $(s_n, \lVert \cdot \rVert_n)$ an adequate family of weighted $\ell_2$-spaces, I guess the inclusion $(s,\lVert \cdot \rVert) \hookrightarrow s'$ is continuous if there exists $n$ such that $\lVert \cdot \rVert_n \leq C \lVert \cdot \rVert$? $\endgroup$ – Goulifet Mar 15 '17 at 16:50
  • $\begingroup$ For the seminorm, I am mostly interested by the following case. Let's say that $\mathrm{L}$ is an operator that continuously map $s$ to $\mathcal{X}$ (still the completion of $s$ for the norm $\lVert \cdot \rVert$), and let's assume that $s \subseteq \mathcal{X} \subseteq s'$, then can we identify the completion of $(s ,\lVert \mathrm{L} \cdot \rVert)$ with a subspace embedded in $s'$? $\endgroup$ – Goulifet Mar 15 '17 at 16:55

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