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?