A set is a subspace if and only if all of the following conditions are satisfied:
It is contained in or equal to a vector space.
It is non-empty
It is closed under multiplication by a scalar
It is closed under addition.
The first 2 of these are generally very easy to prove. For the 3rd use the fact that it is non-empty and assume that u is in the set and prove that this implies ku is in the set for any scalar k. For the third, assume that u and v are in the set and show that this implies u+v is in the set.