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Post Closed as "no longer relevant" by S. Carnahan
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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?

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