Exponential Function on Complex Banach Algebra

The papers [23], [24], [4], [5], [2], [20], [21], [9], [1], [22], [13], [15], [16], [12], [10], [11], [17], [14], [25], [3], [7], [6], [19], and [8] provide the notation and terminology for this paper. For simplicity, we adopt the following convention: X denotes a complex Banach algebra, w, z, z1, z2 denote elements of X, k, l, m, n denote natural numbers, s1, s2, s3, s, s ′ denote sequences of X, and r1 denotes a sequence of real numbers. Let X be a non empty normed complex algebra structure and let x, y be elements of X. We say that x, y are commutative if and only if: (Def. 1) x · y = y · x. Let us note that the predicate x, y are commutative is symmetric. One can prove the following propositions: (1) If s2 is convergent and s3 is convergent and lim(s2 − s3) = 0X , then lim s2 = lim s3. (2) For every z such that for every natural number n holds s(n) = z holds lim s = z. (3) If s is convergent and s is convergent, then s · s is convergent. (4) If s is convergent, then z · s is convergent. (5) If s is convergent, then s · z is convergent. (6) If s is convergent, then lim(z · s) = z · lim s. (7) If s is convergent, then lim(s · z) = lim s · z.