Holomorphic functions on 
uncountable
Abstract
In this article we show that 
, the (Fréchet) holomorphic functions on 
, is complete with respect to the topologies 
and 
. The same result for countable I is well known (see [2]) since in this case is a Fréchet space. The extension to uncountable I requires a different approach.For the compact open topology 
we use induction to reduce the problem to the countable case.Next we use the result for to reduce the problem for 
and 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 and . We refer to [2] for background information.
, the (Fréchet) holomorphic functions on , is complete with respect to the topologies and . The same result for countable I is well known (see [2]) since in this case is a Fréchet space. The extension to uncountable I requires a different approach.For the compact open topology we use induction to reduce the problem to the countable case.Next we use the result for to reduce the problem for and 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 and . We refer to [2] for background information.DOI Code:
10.1285/i15900932v10supn1p65
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