Let be a realvector space (e.g., the real continuous functions on a closed
interval ,
two-dimensional Euclidean space , the twice differentiable real functions on , etc.). Then is a real subspace of if
is a subset of and, for every ,
and (the reals),
and . Let be a homogeneous system of linear equations in , ..., . Then the subset of
which consists of all solutions of the system is a subspace of .

More generally, let
be a field with , where is prime, and let denote the -dimensional vector space over
. The number of -D linear subspaces of is

(1)

where this is the q-binomial coefficient
(Aigner 1979, Exton 1983). The asymptotic limit is