Analytic functions in Banach spaces

Abstract

Introduction. In [1] and [5] differential calculus is developed for (real or complex) Banach spaces in a dimension-free manner with scarcely more ado than in the 1-dimensional case. Our purpose is to demonstrate that a similar treatment, without reference to dimension, is available for analytic functions (real or complex Banach spaces) with equal simplicity. In case the domain or range is Rn or Cn the theory comprehends the classic one; but we need no polycylinders. We shall not write a treatise, but merely set up the basic definitions, derive a few classic theorems, and mention one or two points of caution. The reader will find further similar generalizations and details easy (not the fundamental theorem of algebra, to be sure) using some of the standard techniques of our references. We suppose more or less familiarity with those references and we shall use the notation of [I] and [5]. Cl A will denote the closure of A; Int A the interior of A. CnorCn(D: Y), n = 1, * . ., oo, a, denotes the class of maps with domain D and image in Y of continuous differentiability class n=1,***, oo, and n =a means analytic. X and Y denote Banach spaces, D an open set in X. Nr(x) stands for the open ball, center radius r. A diffeomorphism f: D-*f(D) C Y is a homeomorphism of class Cn, with f'(x) a topological isomorphism for each x (in D). We shall obtain, among other results, the inverse and implicit function theorems for analytic functions without reference to dimension or scalar field. Let xi, , xE,-X and an a continuous, symmetric, n-linear map of Xn into Y, i.e. an(Ln(X: Y). In an(xl, * * ., xn), we can restrict the xi to the diagonal, i.e. to be equal; an(x, * * *, x) will be abbreviated anX A power series in x with values in Y is a series of the form n-o a x, where ao is a point of Y. A power series with only a finite number of terms is a polynomial, O akx If an $0, then the polynomial has degree n. (The function an is 0 iff it vanishes identically on the diagonal, since the nth derivative Ofan Xn is n!a..)