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. Vector-valued holomorphic functions defined in the complex plane 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 Moreover, the composition T o f : U → C is a complex-valued holomorphic function. Therefore, for γ a simple closed curve whose interior is contained in U, the integral on the right is zero, by the classical Cauchy integral theorem. Then, since T is arbitrary, it follows from the Hahn–Banach theorem that which proves the Cauchy integral theorem in the vector-valued case. 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. A useful criterion for a function f : U → X to be holomorphic is that T o f : U → C is a holomorphic complex-valued function for every continuous linear functional T : X → C. Such an f is weakly holomorphic. It can be shown that