  Intersection numbers for subspace designs are introduced and $q$-analogs of
the Mendelsohn and K\"ohler equations are given. As an application, we are able
to determine the intersection structure of a putative $q$-analog of the Fano
plane for any prime power $q$. It is shown that its existence implies the
existence of a $2$-$(7,3,q^4)_q$ subspace design. Furthermore, several
simplified or alternative proofs concerning intersection numbers of ordinary
block designs are discussed.
