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.