In wikipedia, distribution and locally convex topological space. It says that for set of all smooth functions on a manifold, we can use completion to construct seminorms $p_{a, b} = sup_x |x^a D_b f(x)|$. For set of smooth functions with compact support $U$, we can use seminorms $p_m = sup_{k \leq m} sup_{x \in U} |D^k f(x)|$. The wikipedia says that it is locally convex topological space. (Though I learned these in order to understanding derivative of delta function as distribution)
My question is that:
Given these seminorms, how can we pick basis of these two sets of smooth functions? (Well, yes, we need to pick module over some ring, but I am neither unsure of which ring we pick and which module we use.)