I have the following question:
In $S(\mathbb{R}^n)$, and for $\alpha, \beta\in \mathbb{N}^n $, we defined \begin{align} ρ_{\alpha,\beta}(\phi) = \sup_{x\in\mathbb{R}^n}|x^{\alpha}D^{\beta}\phi(x)|. \end{align} We defined a distance as follows: we take a bijection $\sigma: \mathbb{N}\mapsto \mathbb{N}^n\times \mathbb{N}^n$, and set \begin{align} d(\phi,\varphi) = \sum_n 2^{-n}\frac{ρ_{\sigma(n)}(\phi-\varphi)}{1 + ρ_{\sigma(n)}(\phi − \varphi)} \end{align} This distance induces a topology in $S(\mathbb{R}^n)$. Starting from these definitions, prove that the family of sets \begin{align} N_{\epsilon, l, m} = \Big\lbrace \phi\in S\mid \sum_{|\alpha|\leq l, |\beta|\leq m}p_{\alpha, \beta}(\phi)<\epsilon\Big\rbrace \end{align} defines a basis of neighborhoods of $0 \in S$.
I know that what we have to do is to assume that we have a open set $U$ in $S(\mathbb{R}^n)$ containing zero, that is, for every $x\in U$ there exists a disc
\begin{align}
B_{r,x}=\lbrace \phi\in S\mid d(x,\phi)<r\rbrace
\end{align}
such that $x\in B_{r,x}\subseteq U$. Then we have to look if there exist some $N_{\epsilon,l,m}$ that defines a basis for $U$. Thats the point I get lost...
I appreciate any help :)