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!
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 ${\cal S}(\mathbb{R})$.