  We discuss a multiplicative counterpart of Freiman's $3k-4$ theorem in the
context of a function field $F$ over an algebraically closed field $K$. Such a
theorem would give a precise description of subspaces $S$, such that the space
$S^2$ spanned by products of elements of $S$ satisfies $\dim S^2 \leq 3 \dim
S-4$. We make a step in this direction by giving a complete characterisation of
spaces $S$ such that $\dim S^2 = 2 \dim S$. We show that, up to multiplication
by a constant field element, such a space $S$ is included in a function field
of genus $0$ or $1$. In particular if the genus is $1$ then this space is a
Riemann-Roch space.
