If we have a vector space $V$ this has a dual $V^*$. When a learned about dual spaces we were told it has the same dimension as the vector space and I think the lecturer said they were unique (if not the notation of just using * seems a bit vague).
Then I noticed something. Say $V$ is a function space, with a set of basis functions $e_i (x)$. Then I can create a range of dual vector spaces:
1) $$ \int e_i(x) w_1 (x) ● $$
2) $$ \int e_i(x) w_2(x) ● $$
Where $w_1\ne w_2$. We now either have 2 dual spaces or the dual space is twice as big as the origional space.