On the product of vector spaces in a commutative field extension

Let $K \subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace $<AB>$ spanned by the product set $AB=\{ab \mid a \in A, b \in B\}$. If $\dim_K A = r$ and $\dim_K B = s$, how small can the dimension of $<AB>$ be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on $\dim_K < AB>$ turns out, in this case, to be provided by the numerical function $$\kappa_{K,L}(r,s) = \min_{h} (\lceil r/h\rceil + \lceil s/h\rceil -1)h,$$ where $h$ runs over the set of $K$-dimensions of all finite-dimensional intermediate fields $K \subset H \subset L$. This bound is closely related to one appearing in additive number theory.