# Properties of the Schwartz space

I uploaded a photo of my problem because latex isn't working very well here.

Note that it's a bit of awkward notation to use the index $$n$$ for both the sequence, and the definition of $$d_{\mathcal{S}}$$. I'll use $$k$$ for the latter
1. If $$p_{\alpha, \beta}(f_n-f)\to 0$$ for all $$\alpha,\beta$$. Fix $$\varepsilon>0$$ and pick first $$K$$ such that $$\sum_{k=K+1}^\infty c_k <\varepsilon/2$$, then pick $$N=N(K,\varepsilon)$$ such that $$\sum_{k=1}^K p_{\alpha_k,\beta_k}(f_n-f)<\varepsilon/2$$.
2. Use the "trivial" bound $$\| \mathcal{F} F\|_\infty \leq \| F\|_1$$, so that $$p_{\alpha,\beta}(\hat{f}_n-\hat{f})\lesssim \| \partial^{\alpha}(x^\beta(f_n-f)(x))\|_1,$$ and now use the product rule to bound the right hand side in terms of a finite sum of terms of the form $$\| x^{\beta'}\partial^{\alpha}(f_n-f)(x)\|_{1},$$ with $$\beta'\leq \beta$$. Now use that $$f_n-f$$ is Schwartz again to get, for instance, that $$|x^{\beta'}\partial^\alpha(f_n-f)(x)|\lesssim p_{\alpha'', \beta''}(f_n-f)(1+|x|)^{-2}, \qquad x\in \mathbb{R}^n,$$ for some $$\alpha'', \beta''$$. This gives the integrability and continuity.