I'm starting to study distributions and on the lecture notes I'm reading the author defines a test-function as a function $f : U\subset \mathbb{R}^n\to \mathbb{R}$ which is infinitely differentiable and has compact support. He denotes the set of test-functions on $U$ by $\mathcal{D}(U)$.
It is obvious then that the set of test-functions carries a natural structure of a vector-space when we consider the usual pointwise addition and multiplication by scalar.
My question is: this space $\mathcal{D}(U)$ carries any other natural structure like that of a metric or normed vector space? Or for the purposes of distribution theory it is just always treated as a vector space without any topological or metric notions defined on it?