In calculus, there was a difference between functions of a real variable and functions of a
complex variable. In the latter case the existence of a single derivative implied the
existence of all derivatives and in fact the Taylor series converged to the function. It is
reasonable to ask if a similar phenomenon occurs in the case of complex Banach spaces
versus real Banach spaces. This section presents a quick introduction to this topic
based on the assumption that the reader has had some exposure to complex
analysis. Some of the details involving questions of convergence and term by term
differentiation are left to the reader. Also if h maps an open subset of ℂ to a complex
Banach space X, and has a first derivative, then the usual Cauchy integral
formula,
∫
h(z) =-1- h(w)dw,
2πi C w− z
holds if C is a circle contained, together with its interior, in the open set on which h has
a derivative. The integral can be defined as the ordinary Riemann integral using Riemann
sums or it can be defined in terms of a Bochner integral. These details are routine and
are left to the reader. There are several equivalent definitions of an analytic function
defined on a complex Banach space. The following is the one we will use since it
resembles the familiar definition encountered in undergraduate complex variable
courses.
Definition 20.13.1Let X and Y be complex Banach spaces and let U ⊆ X be an openset. We say f : U → Y is analytic and bounded on U if
z → f (x+ zh) is analytic for x ∈ U,h ∈ X and |z| small enough
exists for all x ∈ U and also
||f (x)||
≤ M < ∞ for all x ∈ U. Here z ∈ ℂ andx,h ∈ X.
Let h ∈X^{l} and consider all z ∈ℂ^{l} with
∥z∥
_{ℂl}≡ max
(|zm|,m = 1,⋅⋅⋅,l)
sufficiently
small. Let C_{1} be a sufficiently small circle centered at 0. Then consider
( )
∑l
zm → f x + zmhm
m=1
which is analytic on and inside C_{1}. Thus using the Cauchy integral formula,