I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.


Here $\Omega$ is any open subset of some $\mathbb{R}^n$ and $C^\infty(\Omega)$ is the space of complex-valued $C^\infty$ functions on $\Omega$. ${C^\infty}'(\Omega)$ is the dual space of $C^\infty(\Omega)$.


So, my questions are:


1. What topology is given on $C^\infty(\Omega)$ in general? Here $\Omega$ doesn't have to be bounded at all.. We need to deal with this issue in order to define ${C^\infty}'(\Omega)$.
2. What topology is given on the dual space ${C^\infty}'(\Omega)$ then? This is necessary to define the notion of "$C^\infty$" for the functions whose domain is ${C^\infty}'(\Omega)$.


Could anyone clarify?



How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?
The standard topology on $C^\infty(\Omega)$ is that of uniform convergence on compact subsets of $\Omega$ of all derivatives which is given by the seminorms $\|f\|\_{K,n}=\sup\{|\partial^\alpha f(x)|: x\in K, |\alpha|\le n\}$ with $K\subseteq \Omega$ compact and $n\in \mathbb N$. This topology makes $C^\infty(\Omega)$ a nuclear Fréchet space. The standard topology on the dual is the topology of uniform convergence on all bounded (in this case, this is the same as on all compact) sets. This makes $C^\infty(\Omega)'=\mathscr E'(\Omega)$ (in Schwartz' notation, it is the space of distributions with compact support in $\Omega$) a very nice locally convex space (complete, ultrabornological, nuclear,...).


