Linear Combinations in Real Unitary Space

(1) Let V be a real unitary space, A be a subset of V , and x be a set. Then x ∈ Lin(A) if and only if there exists a linear combination l of A such that x = ∑ l. (2) For every real unitary space V and for every subset A of V and for every set x such that x ∈ A holds x ∈ Lin(A). (3) For every real unitary space V holds Lin( / 0the carrier of V ) = 0V . (4) For every real unitary space V and for every subset A of V such that Lin(A) = 0V holds A = / 0 or A = {0V}. (5) Let V be a real unitary space, W be a strict subspace of V , and A be a subset of V . If A = the carrier of W , then Lin(A) = W. (6) For every strict real unitary space V and for every subset A of V such that A = the carrier of V holds Lin(A) = V. (7) For every real unitary space V and for all subsets A, B of V such that A ⊆ B holds Lin(A) is a subspace of Lin(B).