There exists a finite subset of the carrier of V which is a basis of V . One can verify that there exists a real linear space which is strict and finite dimensional. Let V be a real linear space. There exists a finite subset of V which is a basis of V . A is finite.
V is finite dimensional.
If V is finite dimensional, then W is finite dimensional. Let V be a real linear space. Observe that there exists a subspace of V which is finite dimensional and strict. Let V be a finite dimensional real linear space. Note that every subspace of V is finite dimensional.