Banach space valued analytic functions

The classical notions of complex analytic function, holomorphic functionMathworldPlanetmath and contour integral of a complex function are easily generalized to functions f:X taking values on a complex Banach spaceMathworldPlanetmath X.

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 generalizationPlanetmathPlanetmath.

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.

0.1 Analiticity

Let Ω be an open set and X a complex Banach space.

A function f:ΩX is said to be analytic if each point λ0Ω has a neighborhood in which f is the uniform limit of a power seriesMathworldPlanetmath with coefficients in X centered in λ0


Abel’s theorem on power series is still applicable changing absolute valuesMathworldPlanetmathPlanetmathPlanetmath |.| by vector norms . when appropriate.

0.2 Holomorphicity

A function f:ΩX is said to be differentiableMathworldPlanetmathPlanetmath at a point λ0Ω if the following limit exists (as a limit in X)


f is said to be in SΩ if it is differentiable in a neighborhood of S.

The following Lemma is usefull in the generalization of the classical theory of holomorphic functions.

Lemma 1 - Let f:ΩX be a differentiable function at λ0Ω. Let ϕ:X be a continuousMathworldPlanetmathPlanetmath linear functionalMathworldPlanetmath in X. Then ϕf:Ω is differentiable at λ0 (in the classical sense) and


Proof :

(ϕf)(λ0) = limλλ0ϕ(f(λ))-ϕ(f(λ0))λ-λ0
= limλλ0ϕ(f(λ)-f(λ0)λ-λ0)
= ϕ(limλλ0f(λ)-f(λ0)λ-λ0)
= ϕ(f(λ0))

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.

Let γ:[a,b] be a piecewise smooth path in Ω. Let f:ΩX be a continuous function. Let 𝒫={t0,t1,,tn} be a partitionPlanetmathPlanetmath of [a,b].

We define the


and the of a partition 𝒫 as


The contour integral of f along γ is the element of X defined by


It can be shown that this limit always exists for continuous functions f.

The following Lemma is also usefull

Lemma 2 - Let γ and f be as above. Let ϕ: be a continuous linear functional in X. Then


Proof -

ϕ(γf(λ)𝑑λ) = ϕ(lim𝒫0Rγ(f,𝒫))
= ϕ(lim𝒫0k=1nf(γ(tk))(γ(tk)-γ(tk-1)))
= lim𝒫0ϕ(k=1nf(γ(tk))(γ(tk)-γ(tk-1)))
= lim𝒫0k=1nϕ(f(γ(tk)))(γ(tk)-γ(tk-1))
= γϕf(λ)𝑑λ

0.4 Remarks

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 f:ΩX a continuous function with antiderivative F. Let γ:[a,b]Ω be a piecewise smooth path. Then


Proof : Let ϕ:X be a continuous linear functional. Using Lemmas 1 and 2


(ϕF) is a continuous function Ω. As we know, this theorem is valued for complex valued functions. Then




As X is a Banach space, its dual spaceMathworldPlanetmath X separates points (, so we must have γf(λ)𝑑λ-(F(γ(b))-F(γ(a)))=0 i.e.

Title Banach space valued analytic functions
Canonical name BanachSpaceValuedAnalyticFunctions
Date of creation 2013-03-22 17:29:33
Last modified on 2013-03-22 17:29:33
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Feature
Classification msc 46G20
Classification msc 46G12
Classification msc 46G10
Classification msc 30G30
Classification msc 47A56
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