Algebra of Vector Functions

The terminology and notation used in this paper have been introduced in the following papers: [10], [5], [2], [3], [1], [12], [9], [4], [6], [11], [8], and [7]. For simplicity we adopt the following rules: X, Y will denote sets, C will denote a non-empty set, c will denote an element of C, V will denote a real normed space, f , f1, f2, f3 will denote partial functions from C to the carrier of V , and r, p will denote real numbers. We now define several new functors. Let us consider C, V , f1, f2. The functor f1 + f2 yielding a partial function from C to the carrier of V is defined as follows: (Def.1) dom(f1+f2) = dom f1∩dom f2 and for every c such that c ∈ dom(f1+ f2) holds (f1 + f2)(c) = f1(c) + f2(c). The functor f1 − f2 yields a partial function from C to the carrier of V and is defined as follows: (Def.2) dom(f1−f2) = dom f1∩dom f2 and for every c such that c ∈ dom(f1− f2) holds (f1 − f2)(c) = f1(c)− f2(c). Let us consider C, and let us consider V , and let f1 be a partial function from C to , and let us consider f2. The functor f1 f2 yielding a partial function from C to the carrier of V is defined by: (Def.3) dom(f1 f2) = dom f1 ∩ dom f2 and for every c such that c ∈ dom(f1 f2) holds (f1 f2)(c) = f1(c) · f2(c). Let us consider C, V , f , r. The functor r f yielding a partial function from C to the carrier of V is defined as follows: (Def.4) dom(r f) = dom f and for every c such that c ∈ dom(r f) holds (r f)(c) = r · f(c).