Holomorphic Functions Between Topological Vector Spaces
In the fully general situation, given two topological vector spaces X and Y over the complex numbers and an open set U in 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.
Read more about this topic: Infinite-dimensional Holomorphy
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