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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?

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up vote 15 down vote accepted

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-etalia "Generalized Functions". In the latter, the examples are very small and tangible, but (to my taste) tremendously illuminating about families of distributions.

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I suppose that is what I suspected; thanks. Also, it is also helpful to know that it holds for an even more general situation. I was indeed thinking of the Bochner integral; it didn't even occur to me to use that sort of weak integral actually, so that's a nice heads up. –  Christopher A. Wong Jan 1 '12 at 20:38
    
Also, is there any (modern) text focusing on this general theory of holomorphic functions that is available? It's pretty easy to see that all the standard results hold, but it would be nice to see how far it can be developed, depending on how we add some additional structure to the topological vector space (for example, infinite products? Riemann surfaces?) –  Christopher A. Wong Jan 1 '12 at 20:39

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