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Let $\Omega \subset R^n$ be open, and let us equip $\mathcal E = C^\infty$ with the locally convex topology induced by the following family of semi-norms:

For $K \subset \Omega$ and $\alpha, \beta \in \mathbb N _ 0 ^n$, let $|f|_{K,\alpha,\beta} = sup_{x \in K} | \partial_x^{\alpha} f(x) |$.

It is easy to see that compact sets are closed and bounded within that topology. But how does the opposite direction work?

Thank you!

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