Cauchy Sequence of Complex Unitary Space

The terminology and notation used in this paper are introduced in the following papers: [22], [3], [20], [9], [5], [12], [10], [11], [15], [2], [18], [4], [1], [21], [16], [17], [14], [13], [19], [6], [7], and [8]. For simplicity, we follow the rules: X denotes a complex unitary space, s1, s2, s3 denote sequences of X, R1 denotes a sequence of real numbers, C1, C2, C3 denote complex sequences, z, z1, z2 denote Complexes, r denotes a real number, and k, n, m denote natural numbers. The scheme Rec Func Ex CUS deals with a complex unitary space A, a point B of A, and a binary functor F yielding a point of A, and states that: There exists a function f from N into the carrier of A such that f(0) = B and for every element n of N and for every point x of A such that x = f(n) holds f(n + 1) = F(n, x) for all values of the parameters. Let us consider X, s1. The functor ( ∑ κ α=0(s1)(α))κ∈N yields a sequence of X and is defined as follows: (Def. 1) ( ∑ κ α=0(s1)(α))κ∈N(0) = s1(0) and for every n holds ( ∑ κ α=0(s1)(α))κ∈N(n+ 1) = ( ∑ κ α=0(s1)(α))κ∈N(n) + s1(n + 1). One can prove the following propositions: (1) ( ∑ κ α=0(s2)(α))κ∈N + ( ∑ κ α=0(s3)(α))κ∈N = ( ∑ κ α=0(s2 + s3)(α))κ∈N. (2) ( ∑ κ α=0(s2)(α))κ∈N − ( ∑ κ α=0(s3)(α))κ∈N = ( ∑ κ α=0(s2 − s3)(α))κ∈N. (3) ( ∑ κ α=0(z · s1)(α))κ∈N = z · ( ∑ κ α=0(s1)(α))κ∈N.