Equivalence of norms on Schwarz space Consider the following norms on the Schwarz space, for $1\leq q\leq \infty$
$$\lVert f \rVert_{\alpha,\beta, p}=\lVert x^{\alpha}\partial^\beta f\lVert_{L^p}$$
I want to show that the norms $\lVert \cdot \rVert_{\alpha,\beta, p}$  define the same topology as the standard one on Schwarz space where the topology is given by $$\lVert f \rVert_{\alpha,\beta}=\lVert (1+|x|)^{\alpha}\partial^\beta f\lVert_{u}$$
It suffices to show that either set of seminorms provide a local basis at 0 in the topology generated by the other. 
For one direction we fixed $\alpha', \beta', \epsilon'$, and want to show there is some $\alpha, \beta, \epsilon$ such that $\{\Vert \cdot \rVert_{\alpha,\beta, p}<\epsilon\}\subset$ $\{\Vert \cdot \rVert_{\alpha',\beta'}<\epsilon'\}$.
For this we truncate the integral $\int|f(x)|dx$ by the unit disc and its complement, and then $\{\Vert \cdot \rVert_{0,0, p}<\epsilon\}$ is  in $\{\Vert \cdot \rVert_{0,0}<\epsilon'\}\bigcap \{\Vert \cdot \rVert_{N,0}<\epsilon''\}$ for large enough $N$. Since $x^{\alpha}\partial^\beta f $ is in Schwarz space if $f$ is in Schwarz space, we have $\{\Vert \cdot \rVert_{\alpha,\beta, p}<\epsilon\}\subset$ $\{\Vert \cdot \rVert_{\alpha',\beta'}<\epsilon'\}$.
I don't know how to deal with the other direction. It seems I need to bound sup norm of a Schwarz function $f$ by $L^p$ norm of some $x^\alpha f$.
Thanks for your help.
 A: I suspect the direction you've done gets the inclusion the wrong way around. The estimate you easily get for $\|f\|_{L^p}^p$ is along the lines of
$$\int |f|^p dx = \int_{B(1)} |f|^p dx + \int_{B(1)^c} |f|^p dx \leq \|f\|_{0,0} + \int_{B(1)^c} (1+|x|)^{-N} dx \|f\|_{N,0} =  \|f\|_{0,0} + C \|f\|_{N,0}$$
where we chose $N$ (independent of $f$) large enough that $\int_{B(1)^c} (1+|x|)^{-N} dx < \infty$. This implies that
$$\{\|\cdot\|_{0,0} < \frac{\varepsilon}{2}\} \cap \{\|\cdot\|_{N,0} < \frac{\varepsilon}{2C}\} \subseteq \{\|\cdot\|_{0,0,p} \leq \varepsilon\}.$$ It's then not too hard to replace the subscript $0,0,p$ with $\alpha,\beta,p$ for whatever $\alpha, \beta$ you want.
The other direction requires us to show that $\varepsilon'>0$ and $\alpha',\beta'$ we can find $\varepsilon$ and finitely many pairs $(\alpha,\beta)$ such that
$$\bigcap_{\alpha,\beta} \{\|\cdot \|_{\alpha,\beta,p}<\varepsilon\} \subseteq \{\|\cdot\|_{\alpha',\beta'} < \varepsilon'\}.$$
This means we want to convert finitely many bounds on seminorms of the type $\|\cdot\|_{\alpha,\beta,p}$ into a bound on the seminorm $\|\cdot\|_{\alpha',\beta'}$. One way to do this is via Sobolev embeddings. 
First let's demonstrate the idea in the case $\alpha' = \beta' = 0$. Here I will always take $\alpha = 0$ also. For $k > \frac{n}{p}$ (where $n$ is the dimension of your underlying Euclidean space), $W^{k,p}(\mathbb{R}^n)$ is continuously embedded in $C(\mathbb{R}^n)$ and so $\|\cdot\|_u \leq C \|\cdot\|_{W^{k,p}(\mathbb{R}^n)}$ for some constant $C$. Now
$$\|\cdot\|_{W^{k,p}(\mathbb{R}^n)}^p = \sum_{|\beta| \leq k} \|\cdot\|_{0,\beta,p}^p$$
so let $N = |\{ \beta: |\beta| \leq k\}|$. Then the above shows that
$$\bigcap_{|\beta| \leq k} \{ \|\cdot\|_{0,\beta,p} < \frac{\varepsilon'}{NC}\} \subseteq \{\|\cdot\|_{0,0} < \varepsilon'\}.$$
For general $\alpha',\beta'$, the difficult term to control is $\|x^{\alpha'}\partial^{\beta'}f\|_u$. We can do this using the above method as long as we have control on $\|\partial^{\gamma}(x^{\alpha'}\partial^{\beta'}f)\|_{L^p}$ for a suitable collection of multi-indices $\gamma$. This will follow from control on a finite family of the seminorms $\|\cdot\|_{\alpha,\beta,p}$ by the Leibniz rule.
