One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust Omega $\Omega$ by compacts $(K_n)$ and to take the metric induced by the semi-norm system $$ {\| f \|} _ {n} := \| f \|_{C^n(K _ {n} )}, $$ i.e. $$ d(f, g) = \sum _ n 2^{-n} \frac{ \|f-g\| _ {n} }{ 1+\|f-g\| _ {n} } $$ I read (without any reference) that this yields a non-complete space.

Do you know a reference or a concrete example how to show non-completeness?

One naive way to define a topology on test functions would be to exhaust Omega by compacts $(K_n)$ and to take the metric induced by the seminorm semi-norm system

\| $$ {\| f \|n |} _ {n} := \| f \||_{C^n(K _ {C^n(K_n)}, n} )}, $$ i.e. $$ d(f, g) = \sum_n sum _ n 2^{-n} \frac{ \|f-g\|_n |f-g\| _ {n} }{ 1+\|f-g\|_n 1+\|f-g\| _ {n} } $$

One naive way to define a topology on test functions would be to exhaust Omega by compacts $(K_n)$ and to take the metric induced by the seminorm system

\| f \|n := \| f \|{C^n(K_n)}, i.e.

d(f, g) = \sum_n 2^{-n} \frac{ \|f-g\|_n }{ 1+\|f-g\|_n }

I read (without any reference) that this yields a non-complete space.

Do you know a reference or a concrete example how to show non-completeness?