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 $\lambda_{0}$

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

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 :

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

and the norm of a partition $\mathcal{P}$ as

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 -

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

$(\phi\circ F)^{{\prime}}$ 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 $X^{{\prime}}$ separates points, so we must have $\int_{{\gamma}}f(\lambda)d\lambda-(F(\gamma(b))-F(\gamma(a)))=0$ i.e.