For 'H' to be a subspace of a vector space 'V' it must have 3 properties:
 The zero vector of V is in H.
 For each 'u' and 'v' in H then v+u is also in H (closed under +).
 For each 'v' in H and scalar 's' then s*v is also in H (closed under scalar multipication).
Example of a Subspace
V 

a vector space in 3 dimensions x,y and z  
H 

a subspace of V formed by setting the z dimenstion to zero. 
Theorm
If v_{1}…v_{p} are in a vector space V, then span{v_{1}…v_{p}}is a subspace of V.
Lattice of Vector Subspaces
We can represent the relationships between a whole set of subspaces by a lattice structure (lattices are described on this page).
One way to encode this lattice structure of algebras is, as a Clifford algebra, as explained on this page.