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I would like to prove that two given norms in the space of smooth functions are equivalent in an open set, is it enough to show that they are equivalent for any compactly contained open set? why?

Edit: Clarification promoted from the comments: Consider the space $ \{f\colon U \to \mathbb{R}:f \text{ is smooth}\} $ and consider two given norms on this space. Is it enough in order to prove that these two norms are equivalent that in any of the spaces $ \{f\colon V \to \mathbb{R} : f \text{ is smooth}\} $ where $V$ is compactly contained in $U$, the two norms are equivalent?

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    @Jochen I don't think it is important since with cut off functions you can extend the functions defined on $V$ to the whole domain $U$.2012-09-11

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