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One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust $\Omega$ 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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closed as no longer relevant by S. Carnahan Jan 8 '11 at 13:40

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Oups, the answer is simple. Take $f_n$ such that $0 \le f_n \le 1$ with $f_n = 1$ on $K_n$ but $f_n = 0$ outside $K_{n+1}$. Then $\| f_n-f_m \|_k = 0$ for all $k \le \min(n, m)$. Therefore, letting $N$ such that $\sum_N^\infy 2^{-j} < \epsilon$ yields $d(f_n, f_m) < \eps$ for all $n, m > N$. – Bernhard Jan 8 '11 at 12:47
Do you wish your question to be closed? – Wadim Zudilin Jan 8 '11 at 12:55