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

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

So, my questions are:

1. What topology is given on $E(\Omega)$ in general? Here $\Omega$ doesn't have to be bounded at all.. We need to deal with this issue in order to define $E'(\Omega)$.

2. What topology is given on the dual space $E'(\Omega)$ then? This is necessary to define the notion of "$C^\infty$" for the functions whose domain is $E'(\Omega)$.

Could anyone clarify?

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?