Let be $\Omega \subset \mathbb R^n$ and open set.

The space of smooth function $C^\infty (\Omega)$ is endowed with the topology generated by the seminorms

$\{p_{K,m}: \;K \subset \Omega \;\text{compact}, m \in \mathbb N \}$ where $p_{K,m}(u)= \max_{|\alpha|\le m} \sup_{x \in K} |\partial^\alpha u(x)|$

The same topology can be construct choosing appropriate compact subsets $K_1 \subset K_2 \subset ... \subset \Omega$, so that $C^\infty (\Omega)$ results to be a Frechet space.

I consider the linear subspace

$C^\infty _H(\Omega)=\{u \in C^\infty (\Omega): \;\text{supp} \;u \subset H \}$ where $H \subset \Omega$ is compact

endowed with the induced topology.

Let $K \subset \Omega$ be compact. If $H \subseteq K$ for every $u$ with support in $H$ one has $p_{K,m}(u)=p_{H,m}(u)$; if $K \subset H$ for every $u$ as above one has $p_{K,m}(u) \le p_{H,m}(u)$.

So the topology induced by $C^\infty (\Omega)$ on $C^\infty _H(\Omega)$ coincide with the topology generated by the family of seminorms $\{p_{K,m}: \; m \in \mathbb N \}$

Question: Is the last conclusion correct?


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