For a Banach space $V$ over $\mathbb{C}$ and $U \subset \mathbb{C}$ open, one can easily check that the notions of holomorphy hold for maps $f: U \rightarrow V$ just as in the classical sense. Indeed, I believe one can even use the Lebesgue integral for contour integration since Lebesgue integration on Banach-space-valued functions is also well-developed.
It seems that almost all the major results of classical complex analysis for holomorphic functions $f: U \rightarrow \mathbb{C}$ still hold in an analogous manner. Where are some crucial points where the theory differs when $f$ takes values in a Banach space?
Best Answer
L. Schwartz and A. Grothendieck made clear, by very early 1950s, that the Cauchy (-Goursat) theory of holomorphic functions of a single complex variable extended with essentially no change to functions with values in a locally convex, quasi-complete topological vector space. Cauchy integral formulas, residues, Laurent expansions, etc., all succeed (with trivial modifications occasionally).
Conceivably one needs a little care about the notion of "integral". The Gelfand-Pettis "weak" integral suffices, but/and a Bochner version of "strong" integral is also available.
Further, in great generality, as Grothendieck made clear, "weak holomorphy" (that is, $\lambda\circ f$ holomorphic for all (continuous) linear functionals $\lambda$ on the TVS) implies ("strong") holomorphy (i.e., of the TVS-valued $f$).
(Several aspects of this, and supporting matter, are on-line at http://www.math.umn.edu/~garrett/m/fun/Notes/09_vv_holo.pdf and other notes nearby on http://www.math.umn.edu/~garrett/m/fun/)
Edit: in response to @Christopher A. Wong's further question... I've not made much of a survey of recent texts to see whether holomorphic TVS-valued functions are much discussed, but I would suspect that the main mention occurs in the setting of resolvents of operators on Hilbert and Banach spaces, abstracted just a little in abstract discussions of $C^*$ algebras. (Rudin's "Functional Analysis" mentions weak integrals and weak/strong holomorphy and then doesn't use them much, for example.) Schwartz' original book did treat such things, and was the implied context for the first volume of the Gelfand-Graev-etal "Generalized Functions". In the latter, the examples are very small and tangible, but (to my taste) tremendously illuminating about families of distributions.
Edit-edit: @barto's further question is about the behavior of holomorphic operator-valued $f(z)$ at an isolated point $z_o$ where $f(z)$ fails to be invertible. I do not claim to have a definitive answer to this, but only to suggest that the answer may be complicated, since already for the case $f(z)=(T-z)^{-1}$ for bounded, self-adjoint $T$, it seems to take a bit of work (the spectral theorem) to prove that isolated singularities are in the discrete/point spectrum of $T$. But this may be overkill, anyway...