I'm following some notes on functional analysis where it's shown the result presented on the title.
The proofs in the notes is not clear to me.
Before showing the proof I give some context.
The topology on Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is generated by the family of seminorms defined as follow $$
p_{\alpha,m}:= \sup_{ x \in \mathbb{R}^{n} } (1+|x|^{m})|\partial^{\alpha}\varphi(x)|
$$
with $\alpha,m \in \mathbb{N}_0^n$.
Another family of seminorms which generates the same topology is the following
$$
q_{\alpha,Q}(\varphi):=\sup_{ x \in \mathbb{R}^{n} } |Q(x)\partial^{\alpha}\varphi(x)|
$$
where $Q$ is a polynomial function on $\mathbb{R}^n$.
Back to the proof: Leiniz’ rule applied to $\partial^\beta(x^\alpha\varphi(x))$ yields an upper bound for $p_{\beta,m}(x^\alpha\varphi)$ in the form of a finite sum with terms $q_{\beta,Q_{m+|\alpha|−l}}\ (0 \leq l \leq \max(|\beta|, m + |\alpha|))$, where $Q_{m+|\alpha|−l}$ is a polynomial function of order at most $m + |\alpha| − l$.
Doubt on the proof: I get the aim of the proof i.e. showing that $p_{\beta,m}(x^\alpha\varphi)$ can be bounded by a finite sum with terms $q_{\beta,Q_{m+|\alpha|−l}}$, but I can't understand how he uses the Leiniz’ rule to prove that.
I really appreciate any kind of help :)
Let's say that the Leibniz rule wasn't completly clear to me :)
$$ \begin{align} p_{\beta,m}(x^\alpha\varphi)&=\sup_{x\in \mathbb{R}^n} (1+|x|^m)|\partial^\beta(x^\alpha \varphi(x))|\\ &=\sup_{x\in \mathbb{R}^n} (1+|x|^m)|\sum_{\gamma:\gamma\leq\beta}|\binom{\beta}{\gamma} (\partial^{\beta-\gamma} x^\alpha)(\partial^\gamma\varphi(x))| & \text{Leibniz rule}\\ &=\sup_{x\in \mathbb{R}^n} \sum_{\gamma:\gamma\leq\beta} |Q^\gamma_{m+|\alpha|}(\partial^\gamma\varphi(x))|\\ &\leq\sum_{\gamma:\gamma\leq\beta}\sup_{x\in \mathbb{R}^n} |Q^\gamma_{m+|\alpha|}(\partial^\gamma\varphi(x))|\\ &=\sum_{\gamma:\gamma\leq\beta} q_{\gamma,Q^\gamma_{m+|\alpha|}}(\varphi) \leq C\max_{\gamma:\gamma\leq\beta}q_{\gamma,Q^\gamma_{m+|\alpha|}}(\varphi) \end{align} $$ where $C=|\{\gamma: \gamma \leq \beta\}|< \infty$ and $Q^\cdot_{k}$ a polynomial function of order at most $k$.