Complex Valued Functions Space

We adopt the following convention: x1, x2, z are sets, A is a non empty set, and f , g, h are elements of CA. Let us consider A. The functor +CA yielding a binary operation on CA is defined by: (Def. 1) For all elements f , g of CA holds +CA(f, g) = (+C)◦(f, g). Let us consider A. The functor ·CA yielding a binary operation on CA is defined as follows: (Def. 2) For all elements f , g of CA holds ·CA(f, g) = (·C)◦(f, g). Let us consider A. The functor ·CA yielding a function from [:C, CA :] into CA is defined by: (Def. 3) For every complex number z and for every element f of CA and for every element x of A holds ·CA(〈z, f〉)(x) = z · f(x). Let us consider A. The functor 0CA yielding an element of CA is defined by: (Def. 4) 0CA = A 7−→ 0C. Let us consider A. The functor 1CA yields an element of CA and is defined by: (Def. 5) 1CA = A 7−→ 1C.