Characterization of Analytic Functions in Banach Spaces

Abstract

The concept of analyticity may be extended in various ways to functions from one complex Banach space to another. We may ask that the function be differentiable on one-dimensional (complex) subspaces; here one is led to the theory of the GAteaux differential. Or we may prescribe a seemingly much more powerful condition, namely, that the function possesses a development into (abstract) power series about each point of the domain of definition. Here the Fr6chet differential plays a decisive role. The characterization theorem which we are going to derive will serve to show that the functions which fall under the first definition but' not under the second are, from a certain point of view, to be considered as freaks, counter examples rather than examples. They are similar in character to, say, additive functions of a real variable which are not linear. For it turns out that only a very weak continuity property has to be added to the existence of the Gateaux differential in order to ensure the existence of the power series development called for by the second definition. This property-which we shall call B-continuity (and not, as we might, property of Baire)-aniounts to this: if a suitable set of the first category is dropped from the domain of definition, the remaining partial function shall be a continuous one. This property is preserved under passage to the limit and thus shared by large classes of functions. As a byproduct of the theory we obtain the solution of a problem which has long been unsolved. 0. We shall borrow freely from the theory of functions of a topological variable and occasionally from classical function theory. At a critical point we have to bring in the theory of abstract polynomials, and from the theory of Banach spaces and Banach space functions of one complex variable the following theorems will have to be used. (Unless there is explicit notice to the contrary we use only complex Banach spaces.) (0.1) Let F be a subset of a Banach space Y, and Y* the B-space of bounded linear functionals y* on Y to the complex numbers. If every functional y*(y) is bounded on F, the set F is bounded. (In the sense that the norms of its elements form a bounded set.) (0.2) Let f(v) be a function defined on an open set A of complex numbers, with values in a B-space Y.* If for every functional y* the numerical function y*[f(v)] is differentiable on A, the function f(v) is differentiable on A; its derivative is likewise differentiable. (0.3) If f(v) is differentiable for i v I < p, we have for these values of the argument the unique Taylor development f(v) = w [f (n)()]o/n!?. From the principle of the maximum awe need only the part: (0.4) If the function f(v) is differentiable for I v i _ p, then suplr.-p I f(0) I = supwrl=! If'v) 11 585