Dimension of Real Unitary Space

One can prove the following two propositions: (1) Let V be a real unitary space, A, B be finite subsets of V , and v be a vector of V . Suppose v ∈ Lin(A∪B) and v / ∈ Lin(B). Then there exists a vector w of V such that w ∈ A and w ∈ Lin(((A ∪ B) \ {w}) ∪ {v}). (2) Let V be a real unitary space and A, B be finite subsets of V . Suppose the unitary space structure of V = Lin(A) and B is linearly independent. Then B ¬ A and there exists a finite subset C of V such that C ⊆ A and C = A − B and the unitary space structure of V = Lin(B ∪ C). Let V be a real unitary space. We say that V is finite dimensional if and only if: (Def. 1) There exists a finite subset of the carrier of V which is a basis of V .