I am asking myself the following question:
Are space of test functions $\mathcal D$ and the space of distributions $\mathcal D'$ normed spaces (or even Banach spaces)?
My thought. I think that the answer is YES since I intuitively see that it is possible to define the norms in $\mathcal D'$. For examples, let $T\in \mathcal D'$, we may define the norm for $T$ as follows
$$||T||_{\mathcal D'}= \sup_{f\in \mathcal D: ||f||_{\infty} \leq 1} |T(f)|$$
or
$$||T||_{\mathcal D'}= \sup_{f\in \mathcal D: ||f||_{L^2} \leq 1} |T(f)|.$$
However, when I google I see that many papers try to put $\mathcal D'$ and $\mathcal D$ in some bigger Banach spaces than themselves, see e.g. this paper.
So I am really confusing. Am I right?