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What mean a $0$ in the set of functions infinitely differentiable $C_0^\infty$

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If i understood you question, its means the set of $C^{\infty}$ functions with compact support. – Tomás Oct 24 '12 at 13:03
up vote 2 down vote accepted

It means "compactly supported", i.e. elements of that space identically vanish outside a compact subset.

Sometimes it means that the elements $u$ decay to zero "at infinity", in the sense that for any $\epsilon >0$ there exists a compact subset $K$ such that $|u|<\epsilon$ on $\complement K$.

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