  We are interested in characterising pairs $S,T$ of $F$-linear subspaces in a
field extension $L/F$ such that the linear span $ST$ of the set of products of
elements of $S$ and of elements of $T$ has small dimension. Our central result
is a linear analogue of Vosper's Theorem, which gives the structure of vector
spaces $S, T$ in a prime extension $L$ of a finite field $F$ for which
$\dim_FST =\dim_F S+\dim_F T-1,$ when $\dim_F S, \dim_F T\ge 2$ and $\dim_F
ST\le [L:F]-2$.
