Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out seminorms). Let $\alpha=(\alpha_1,\ldots,\alpha_n)$ ($\alpha_i\in\mathbb{N}_{>0}$). Define the operator $D^{\alpha}:C^{\infty}(\Omega)\rightarrow C^{\infty}(\Omega)$ by: $$f\mapsto \left(\frac{\partial}{\partial x_1}\right)^{\alpha_{1}}\dots \left(\frac{\partial}{\partial x_n}\right)^{\alpha_n}f$$ The first assertion is that this map is continuous. But how to start? If we can prove that $D^{\alpha}$ is continuous in $0$ then we are done, but how to show that? Questions about continuity are difficult from my point of view -.-
The second statement is that the multiplication of $f\in C^{\infty}(\Omega)$ with a fixed $\varphi\in C^{\infty}(\Omega)$ is continuous. But why is that the case? how to begin?
Thank you for all help and proves :-)
Write $\Omega =\cup_{i=1}^\infty K_i$ where $K_i\subset \Omega$ are compact sets and $K_i\subset K_{i+1}$. For each $n$ and each multi-index $\alpha=(j_1,...,j_n)$, define the family of seminorms: $$\|u\|_{\alpha,j}=\sup\{|D^\alpha u(x)|:\ x\in K_j\} \tag{1}$$
We can enumarate the family of seminorms and induce the a Fréchet topology in $C^\infty (\Omega)$ by considering the metric $$d(u,v)=\sum _{i=1}^\infty 2^{-i}\frac{\|u-v\|_i}{1+\|u-v\|_i},\ \forall u,v\in C^\infty(\Omega)$$
With respect to this toplogy, we have that $u_n\to u$ in $C^\infty(\Omega)$ if and only if $d(u_n,u)\to 0$ and this is equivalently to say that $\|u-v\|_i\to 0$ for each $i=1,2,...$ and this is equivalently to say that $D^\alpha u_n$ converges to $D^\alpha u$ uniformly in each compact set $K_n$, for all multi-index $\alpha$ (also in each compact set contained in $\Omega$).
Now you have a more tractable way do prove your statements.