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I was wondering if the inclusion of the Schwartz space on the $L^{p}$ spaces was continous. I read this question but it only says that there is an inclusion.

Can we prove that it is continous? i.e.: $\forall \left( \varphi_{n} \right)_{n\in\mathbb{N}} \subset \mathcal{S}(\mathbb{R})$, $\forall \varphi \in \mathcal{S}(\mathbb{R})$, $\varphi_{n} \rightarrow \varphi \Rightarrow \| \varphi_{n} - \varphi \|_{p} \rightarrow 0$.

I am specially interested in the case of $p=2$ so a proof in this case will fit for me.

Thank you very much!

uniquesolution
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Manza
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  • Roughly speaking any time $X$ and $Y$ are complete topological vector spaces and $X\subset Y$ then the inclusion is continuous. There may be counterexamples, but finding one takes a little work. In practical terms the proof that $X\subset Y$ is "always" going to contain a proof of continuity - that certainly happens in this case. – David C. Ullrich Aug 19 '19 at 15:39

1 Answers1

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It is true that the question here only says there is an inclusion, but the answer actually writes down explicitly an inequality which on one side has the $L_p$ norm and on the other side has a quantity involving the semi-norms in ${\cal S}(\mathbb{R})$, which proves that the inclusion is indeed continuous with respect to the usual topologies on $L_p$ and ${\cal S}(\mathbb{R})$.

uniquesolution
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