Holomorphic functions on $C^I, I $ uncountable

Abstract

In this article we show that $H( C^I)$, the (Fréchet) holomorphic functions on $C^I$, is complete with respect to the topologies $τ₀, τ_w$ and $τ_δ$. The same result for countable I is well known (see [2]) since in this case $C^I$ is a Fréchet space. The extension to uncountable I requires a different approach.For the compact open topology $τ₀$ we use induction to reduce the problem to the countable case.Next we use the result for $τ₀$ to reduce the problem for $τ_w$ and $τ_δ$ to the case of homogeneous polynomials.Using a method developed for holomorphic functions on nuclear Fréchet spaces with a basis and, once more,the result for the compact open topology we complete the proof for $τ_w$ and $τ_δ$. We refer to [2] for background information.