The Steinitz Theorem and the Dimension of a Real Linear Space

For simplicity, we follow the rules: V denotes a real linear space, W denotes a subspace of V , x denotes a set, n denotes a natural number, v denotes a vector of V , K1, K2 denote linear combinations of V , and X denotes a subset of the carrier of V . We now state a number of propositions: (1) If X is linearly independent and the support of K1 ⊆ X and the support of K2 ⊆ X and ∑ K1 = ∑ K2, then K1 = K2. (2) Let V be a real linear space and A be a subset of V . If A is linearly independent, then there exists a basis I of V such that A ⊆ I. (3) Let L be a linear combination of V and x be a vector of V . Then x ∈ the support of L if and only if there exists v such that x = v and L(v) 6= 0.