a function defined on an open subset of the complex plane with values in a Fréchet space is holomorphic if, and only if, it is weakly holomorphic. Holomorphic functions between Banach spaces 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. It is no longer true however that if a function is defined and holomorphic in a ball, its power series around the center of the ball is convergent in the entire ball; for example, there exist holomorphic functions defined on the entire space which have a finite radius of convergence. Holomorphic functions between topological vector spaces 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. Unlike the finite dimensional setting, when X and Y are infinite dimensional, the properties of holomorphic functions may depend on which definition is chosen. To restrict the number of possibilities we must consider, we shall only discuss holomorphy in the case when X and Y are locally convex. This section presents a list of definitions, proceeding from the weakest notion to the strongest notion. It concludes with a discussion of some theorems relating these definitions when the spaces X and Y satisfy some additional constraints. Gateaux holomorphy Gateaux holomorphy is the direct generalization of weak holomorphy to the fully infinite dimensional setting. Let X and Y be locally convex topological vector spaces, and U ⊂ X an open set. A function f : U → Y is said to be Gâteaux holomorphic if, for every a ∈ U and b ∈ X, and every continuous linear functional φ : Y → C, the function is a holomorphic function of z in a neighborhood of the origin. The collection of Gâteaux holomorphic functions is denoted by HG(U,Y). In the analysis of Gateaux holomorphic functions, any properties of finite-dimensional holomorphic functions hold on finite-dimensional subspaces of X. However, as usual in functional analysis, these properties may not piece together uniformly to yield any corresponding