A subspace is a subset of a vector space that is itself a vector space under the
inherited operations. Let 
be a real vector 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 nonempty subset of 
and, for every 
, 
and 
(the reals), 
and 
. Let 
be a homogeneous linear
system of equations in 
, ..., 
.
Then the subset 
of 
which consists of all solutions of the linear
system of equations 
is a subspace of 
.
More generally, let 
be a finite field of field
order 
, where 
is prime and 
is an integer, and let
denote the 
-dimensional vector
space over 
. The number of 
-dimensional linear subspaces
of 
is
 |
(1)
|
where this is the q-binomial coefficient (Aigner 1979, Exton 1983). The total number of subspaces of all dimensions is
 |
(2)
|
Its asymptotic limit is
![N(F_(q,n))={c_eq^(n^2/4)[1+o(1)] for n even; c_oq^(n^2/4)[1+o(1)] for n odd,](/images/equations/Subspace/NumberedEquation3.svg) |
(3)
|
where
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
(Finch 2003) where 
is a Jacobi theta function and 
is a q-Pochhammer
symbol. The case 
gives the q-analog of the Wallis
formula.
