A function f : U → X, where U ⊂ C is an open subset and X is a complex Banach space, is called holomorphic if it is complex-differentiable; that is, for each point z ∈ U the following limit exists: One may define the line integral of a vector-valued holomorphic function f : U → X along a rectifiable curve γ : [a, b] → U in the same way as for complex-valued holomorphic functions, as the limit of sums of the form where a = t0 < t1 < ... < tn = b is a subdivision of the interval [a, b], as the lengths of the subdivision intervals approach zero.
It is a quick check that the Cauchy integral theorem also holds for vector-valued holomorphic functions.
Using this powerful tool one may then prove Cauchy's integral formula, and, just like in the classical case, that any vector-valued holomorphic function is analytic.
More generally, given two complex Banach spaces X and Y and an open set U ⊂ X, f : U → Y is called holomorphic if the Fréchet derivative of f exists at every point in U.
One can show that, in this more general context, it is still true that a holomorphic function is analytic, that is, it can be locally expanded in a power series.
[1] In general, given two complex topological vector spaces X and Y and an open set U ⊂ X, there are various ways of defining holomorphy of a function f : U → Y.
A function f ∈ HG(U,Y) is holomorphic if, for every x ∈ U, the Taylor series expansion (which is already guaranteed to exist by Gateaux holomorphy) converges and is continuous for y in a neighborhood of 0 ∈ X.