Banach space valued analytic functions
Moreover, the classical theory of complex analytic functions can still be applied, with suitable adjustments, to Banach space valued functions. In this way, important theorems such as Liouville’s theorem remain valid under this generalization.
In this entry we provide the definitions of analyticity and holomorphicity for Banach space valued functions, we give a definition of countour for this type of functions and discuss some useful results which enable the generalization of the classical theory.
Let be an open set and a complex Banach space.
A function is said to be differentiable at a point if the following limit exists (as a limit in )
is said to be in if it is differentiable in a neighborhood of .
The following Lemma is usefull in the generalization of the classical theory of holomorphic functions.
0.3 Contour Integrals
The usual way to relate the theory of complex analytic functions with the theory of holomorphic functions is by the use contour integrals. It is not different for Banach space valued functions.
We will define contour integrals for continuous Banach space valued functions but there’s no particular reason, besides the simplicity of , for restricting to this type of functions.
We define the
and the of a partition as
The contour integral of along is the element of defined by
It can be shown that this limit always exists for continuous functions .
The following Lemma is also usefull
Lemma 2 - Let and be as above. Let be a continuous linear functional in . Then
We have seen how the classical definitions generalize in straightforward way to Banach space valued functions. In fact, as we said before, the whole classical theory remains valid with proper adjustments.
As a example, we will prove a well-known theorem in complex analysis this time for Banach space valued functions.
Theorem - Let a continuous function with antiderivative . Let be a piecewise smooth path. Then
Proof : Let be a continuous linear functional. Using Lemmas 1 and 2
is a continuous function . As we know, this theorem is valued for complex valued functions. Then
As is a Banach space, its dual space separates points (http://planetmath.org/DualSpaceSeparatesPoints), so we must have i.e.
|Title||Banach space valued analytic functions|
|Date of creation||2013-03-22 17:29:33|
|Last modified on||2013-03-22 17:29:33|
|Last modified by||asteroid (17536)|
|Synonym||Banach space valued holomorphic function|
|Synonym||analytic Banach space valued function|
|Synonym||holomorphic Banach space valued function|
|Defines||contour integral of Banach space valued functions|