Size of dual space

Question

Here is a rough intuition: 'the smaller a vector/Banach space the larger its dual space'. For finite dimensions of course, a vector space and its dual are the same 'size', but for instance, in the case of the space of test functions (a very `small' space), we expect a very large dual space of distributions. Unfortunately, I don't have a gut intuition about why this is the case - could someone expand on why this intuition holds?