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I have read definitions in my PDE book as follows: If $M$ is a smooth paracompact manifold, the space of all linear functional on $C^\infty(M) $ is denoted by $\mathcal E'$ and the space of all linear functional on $C_0^\infty(M) $ is denoted by $\mathcal D'$.

I have already known the topologies of these 4 spaces when $M$ is $\mathbb R^n$, can you give me the general description of the topologies of these spaces?

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Usually function spaces have the compact-open topology. en.wikipedia.org/wiki/Compact-open_topology –  Grumpy Parsnip Apr 18 '12 at 1:26
    
Jim, the compact-open topology is much weaker than the topology on distributions. If I recall the compact-open topology can be thought of as uniform convergence on compact sets, while the topology on the space of distributions can be thought of (more or less) as uniform convergence of all derivatives on compact sets. –  Chris Janjigian Apr 18 '12 at 1:32
    
@Chris: thanks for the clarification! –  Grumpy Parsnip Apr 19 '12 at 2:08

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