Complex Linear Space and Complex Normed Space

We consider CLS structures as extensions of loop structure as systems 〈 a carrier, a zero, an addition, an external multiplication 〉, where the carrier is a set, the zero is an element of the carrier, the addition is a binary operation on the carrier, and the external multiplication is a function from [: C, the carrier :] into the carrier. Let us observe that there exists a CLS structure which is non empty. Let V be a CLS structure. A vector of V is an element of V . Let V be a non empty CLS structure, let v be a vector of V , and let z be a Complex. The functor z · v yielding an element of V is defined as follows: (Def. 1) z · v = (the external multiplication of V )(〈z, v〉). Let Z1 be a non empty set, let O be an element of Z1, let F be a binary operation on Z1, and let G be a function from [: C, Z1 :] into Z1. One can verify that 〈Z1, O, F, G〉 is non empty. Let I1 be a non empty CLS structure. We say that I1 is complex linear space-like if and only if the conditions (Def. 2) are satisfied.