## Subspace

Let be a Real Vector Space (e.g., the real continuous functions on a Closed Interval , 2-D 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 -D Vector Space over . The number of -D linear subspaces of is

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

where

(Finch). The case gives the q-Analog of the Wallis Formula.

References

Aigner, M. Combinatorial Theory. New York: Springer-Verlag, 1979.

Exton, H. -Hypergeometric Functions and Applications. New York: Halstead Press, 1983.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/dig/dig.html