Let $G$ be an open subset of $\mathbb R ^n$ and $D(G)$ denotes the set of smooth functions with compact support in $G$.
Consider following families of seminorms,
- For $f\in D(G)$
$||f||_N = \sup \{ | D^{\alpha}f(x)|: x\in G,|\alpha| \leq N \}$ for $N\in \mathbb N$.
- Let $K_n$ be a nested sequence of compact sets which exhaust $G$ then for $f \in D(G)$ define
$\nu_N (f) = \sup \{ | D^{\alpha}f(x)|: x\in K_N,|\alpha| \leq N \}$ for $N\in\mathbb N$.
where $\alpha$ is a multi-index.
Are topologies induced on $D(G)$ by above two families same?
My guess is NO but I am unable to prove it.
EDIT- I am interested in this question because on $C^\infty(G)$ we give topology induced by family 2. In chapter 6 of Rudin’s Functional Analysis, it says family 1 doesn’t give good topology on $D(G)$ (it’s not complete). But Rudin didn’t talk about family 2 even though it’s subspace topology on $D(G)$ as a subset of $C^\infty(G)$.
With the topology from 2, for functions in $C^\infty(\Omega)$, $\lim_n f_n = f$ can be thought as of $D^\alpha f_n \rightarrow D^\alpha f$ uniformly on any compact subsets, and for each $|\alpha| < \infty$.
One can construct $f_n\in \mathcal{D}(\Omega)$ and yet the limit $f\in C^\infty(\Omega)$ does not have compact support.
2 gives a weaker topology than 1 on $\mathcal{D}(\Omega)$, since 1 is "converge uniformly" and 2 is "converge locally uniformly". For example, let $\phi$ have support in $[0,1]$, define $$\psi_m(x) = \phi(x) + \phi(x-1) + \cdots + \phi(x-m)$$ this sequence is Cauchy in 2, but not Cauchy in 1. This also shows that $\mathcal{D}(\Omega)$ with the topology induced by 2 is not complete. (Similar to Rudin's example on page 151.)