Infinite dimensional holomorphy
In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces (or Fréchet spaces more generally), typically of infinite dimension. It is one aspect of nonlinear functional analysis.
A first step in extending the theory of holomorphic functions beyond one complex dimension is considering so-called vector-valued holomorphic functions, which are still defined in the complex plane C, but take values in a Banach space. Such functions are important, for example, in constructing the holomorphic functional calculus for bounded linear operators.
Definition. 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. Indeed, if f : U → X is such a function and T : X → C a bounded linear functional, one can show that
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