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Banach space valued analytic functions (Feature)

The classical notions of complex analytic function, holomorphic function and contour integral of a complex function are easily generalized to functions $ f: \mathbb{C} \longrightarrow X$ taking values on a complex Banach space $ 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 generalization.

In this entry we provide the definitions of analyticity and holomorphicity for Banach space valued functions, we give a definition of countour integral for this type of functions and discuss some useful results which enable the generalization of the classical theory.

Analiticity

Let $ \Omega \subseteq \mathbb{C}$ be an open set and $ X$ a complex Banach space.

A function $ f:\Omega \longrightarrow X$ is said to be analytic if each point $ \lambda_0 \in \Omega$ has a neighborhood in which $ f$ is the uniform limit of a power series with coefficients in $ X$ centered in $ \lambda_0$

$\displaystyle f(\lambda)= \sum_{k=0}^{\infty} a_k (\lambda -\lambda_0)^k, \;\;\;\;\;\;\; a_k \in X $

Abel's theorem on power series is still applicable changing absolute values $ \vert.\vert$ by vector norms $ \Vert.\Vert$ when appropriate.

Holomorphicity

A function $ f:\Omega \longrightarrow X$ is said to be differentiable at a point $ \lambda_0 \in \Omega$ if the following limit exists (as a limit in $ X$)

$\displaystyle f'(\lambda_0):= \lim_{\lambda \rightarrow \lambda_0} \frac{f(\lambda)-f(\lambda_0)}{\lambda - \lambda_0} $

$ f$ is said to be holomorphic in $ S \subset \Omega$ 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 : \Omega \longrightarrow X$ be a differentiable function at $ \lambda_0 \in \Omega$. Let $ \phi: X \longrightarrow \mathbb{C}$ be a continuous linear functional in $ X$. Then $ \phi \circ f : \Omega \longrightarrow \mathbb{C}$ is differentiable at $ \lambda_0$ (in the classical sense) and

$\displaystyle (\phi \circ f)'(\lambda_0) = \phi(f'(\lambda_0)) $

Proof :

$\displaystyle (\phi \circ f)'(\lambda_0)$ $\displaystyle =$ $\displaystyle \lim_{\lambda \rightarrow \lambda_0} \frac{\phi (f(\lambda))- \phi (f(\lambda_0))}{\lambda - \lambda_0}$  
  $\displaystyle =$ $\displaystyle \lim_{\lambda \rightarrow \lambda_0} \phi \left(\frac{f(\lambda)- f(\lambda_0)}{\lambda - \lambda_0}\right)$  
  $\displaystyle =$ $\displaystyle \phi \left(\lim_{\lambda \rightarrow \lambda_0} \frac{f(\lambda)- f(\lambda_0)}{\lambda - \lambda_0}\right)$  
  $\displaystyle =$ $\displaystyle \phi (f'(\lambda_0))\;\;\;\;\square$  

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 presentation, for restricting to this type of functions.

Let $ \gamma : [a,b] \longrightarrow \mathbb{C}$ be a piecewise smooth path in $ \Omega \subseteq \mathbb{C}$. Let $ f : \Omega \longrightarrow X$ be a continuous function. Let $ \mathcal{P} = \{t_0, t_1, \dots, t_n\}$ be a partition of $ [a,b]$.

We define the Riemann sums

$\displaystyle R_{\gamma}(f, \mathcal{P}) := \sum_{k=1}^n f(\gamma(t_k))(\gamma(t_k)-\gamma(t_{k-1})) $
and the norm of a partition $ \mathcal{P}$ as
$\displaystyle \Vert\mathcal{P}\Vert :=\max_k \vert t_k - t_{k-1}\vert $

The contour integral of $ f$ along $ \gamma$ is the element of $ X$ defined by

$\displaystyle \int_{\gamma} f(\lambda) d\lambda := \lim_{\Vert\mathcal{P}\Vert \rightarrow 0} R_{\gamma}(f,\mathcal{P}) $

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

The following Lemma is also usefull

Lemma 2 - Let $ \gamma$ and $ f$ be as above. Let $ \phi : \longrightarrow \mathbb{C}$ be a continuous linear functional in $ X$. Then

$\displaystyle \phi \left(\int_{\gamma} f(\lambda) d\lambda \right) = \int_{\gamma} \phi \circ f (\lambda) d\lambda $

Proof -

$\displaystyle \phi \left(\int_{\gamma} f(\lambda) d\lambda \right)$ $\displaystyle =$ $\displaystyle \phi \left(\lim_{\Vert\mathcal{P}\Vert \rightarrow 0} R_{\gamma}(f,\mathcal{P})\right)$  
  $\displaystyle =$ $\displaystyle \phi \left(\lim_{\Vert\mathcal{P}\Vert \rightarrow 0} \sum_{k=1}^n f(\gamma(t_k))(\gamma(t_k)-\gamma(t_{k-1})) \right)$  
  $\displaystyle =$ $\displaystyle \lim_{\Vert\mathcal{P}\Vert \rightarrow 0} \phi \left(\sum_{k=1}^n f(\gamma(t_k))(\gamma(t_k)-\gamma(t_{k-1}))\right)$  
  $\displaystyle =$ $\displaystyle \lim_{\Vert\mathcal{P}\Vert \rightarrow 0} \sum_{k=1}^n \phi(f(\gamma(t_k)))(\gamma(t_k)-\gamma(t_{k-1}))$  
  $\displaystyle =$ $\displaystyle \int_{\gamma} \phi \circ f(\lambda) d\lambda \;\;\square$  

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 simple example, we will prove a well-known theorem in complex analysis this time for Banach space valued functions.

Theorem - Let $ f: \Omega \longrightarrow X$ a continuous function with antiderivative $ F$. Let $ \gamma : [a,b] \longrightarrow \Omega$ be a piecewise smooth path. Then

$\displaystyle \int_{\gamma} f(\lambda) d\lambda = F(\gamma(b)) - F(\gamma(a)) $

Proof : Let $ \phi : X \longrightarrow \mathbb{C}$ be a continuous linear functional. Using Lemmas 1 and 2

$\displaystyle \phi \left(\int_{\gamma} f(\lambda) d\lambda \right) = \int_{\gam... ...i \circ F'(\lambda) d\lambda = \int_{\gamma} (\phi \circ F)'(\lambda) d\lambda $

$ (\phi \circ F)'$ is a continuous function $ \Omega \longrightarrow \mathbb{C}$. As we know, this theorem is valued for complex valued functions. Then

$\displaystyle \int_{\gamma} (\phi \circ F)'(\lambda) d\lambda = (\phi \circ F)(\gamma (b)) - (\phi \circ F)(\gamma (a)) = \phi [F(\gamma (b)) -F(\gamma (a))] $

Therefore

$\displaystyle \phi \left(\int_{\gamma} f(\lambda) d\lambda - (F(\gamma (b)) -F(\gamma (a))) \right) = 0 \;\;\;\; \forall_{\phi \in X'} $

As $ X$ is a Banach space, its dual space $ X'$ separates points, so we must have $ \int_{\gamma} f(\lambda) d\lambda - (F(\gamma (b)) -F(\gamma (a))) =0$ i.e.

$\displaystyle \int_{\gamma} f(\lambda) d\lambda = F(\gamma (b)) -F(\gamma (a)) \;\;\;\square $



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Other names:  Banach space valued holomorphic function, analytic Banach space valued function, holomorphic Banach space valued function
Also defines:  contour integral of Banach space valued functions
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Cross-references: antiderivative, complex analysis, partition, path, piecewise smooth, proof, linear functional, continuous, vector norms, absolute values, Abel's theorem on power series, coefficients, power series, limit, neighborhood, point, open set, definitions, Liouville's theorem, Banach space, complex, functions, complex function, contour integral, holomorphic function, complex analytic function
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This is version 7 of Banach space valued analytic functions, born on 2007-08-21, modified 2007-10-15.
Object id is 9880, canonical name is BanachSpaceValuedAnalyticFunctions.
Accessed 1242 times total.

Classification:
AMS MSC: 30G30 (Functions of a complex variable :: Generalized function theory :: Other generalizations of analytic functions )
 46G10 (Functional analysis :: Measures, integration, derivative, holomorphy :: Vector-valued measures and integration)
 46G12 (Functional analysis :: Measures, integration, derivative, holomorphy :: Measures and integration on abstract linear spaces)
 46G20 (Functional analysis :: Measures, integration, derivative, holomorphy :: Infinite-dimensional holomorphy)
 47A56 (Operator theory :: General theory of linear operators :: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
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