The Exponential Function on Banach Algebra

The notation and terminology used here are introduced in the following papers: [17], [19], [20], [3], [4], [2], [16], [5], [1], [18], [9], [11], [12], [8], [6], [7], [13], [10], [21], [14], and [15]. For simplicity, we use the following convention: X denotes a Banach algebra, p denotes a real number, 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 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. Next we state a number of 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.