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

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 $\lambda_0$

Abel's theorem on power series is still applicable changing absolute values $|.|$ by vector norms $\|.\|$ 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 $X$ )

$f$ 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

Proof : \begin{eqnarray*} (\phi \circ f)'(\lambda_0) & = & \lim_{\lambda \rightarrow \lambda_0} \frac{\phi (f(\lambda))- \phi (f(\lambda_0))}{\lambda - \lambda_0} \\ & = & \lim_{\lambda \rightarrow \lambda_0} \phi \left(\frac{f(\lambda)- f(\lambda_0)}{\lambda - \lambda_0}\right) \\ & = & \phi \left(\lim_{\lambda \rightarrow \lambda_0} \frac{f(\lambda)- f(\lambda_0)}{\lambda - \lambda_0}\right) \\ & = & \phi (f'(\lambda_0))\;\;\;\;\square \end{eqnarray*}

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

The contour integral of $f$ along $\gamma$ 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 $\gamma$ and $f$ be as above. Let $\phi : \longrightarrow \mathbb{C}$ be a continuous linear functional in $X$ . Then

Proof - \begin{eqnarray*} \phi \left(\int_{\gamma} f(\lambda) d\lambda \right) & = & \phi \left(\lim_{\|\mathcal{P}\| \rightarrow 0} R_{\gamma}(f,\mathcal{P})\right) \\ & = & \phi \left(\lim_{\|\mathcal{P}\| \rightarrow 0} \sum_{k=1}^n f(\gamma(t_k))(\gamma(t_k)-\gamma(t_{k-1})) \right) \\ & = & \lim_{\|\mathcal{P}\| \rightarrow 0} \phi \left(\sum_{k=1}^n f(\gamma(t_k))(\gamma(t_k)-\gamma(t_{k-1}))\right) \\ & = & \lim_{\|\mathcal{P}\| \rightarrow 0} \sum_{k=1}^n \phi(f(\gamma(t_k)))(\gamma(t_k)-\gamma(t_{k-1})) \\ & = & \int_{\gamma} \phi \circ f(\lambda) d\lambda \;\;\square \end{eqnarray*}

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

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

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

Therefore

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.