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I am reading this paper, constructing spaces of functions and distributions with exponential growth on Fréchet nuclear spaces and their dual.

Un théorème de dualité entre espaces de fonctions holomorphes à croissance exponentielle, Journal of Functional Analysis

The proof of Proposition 2 page 7 mentions that some space is Fréchet and reflexive as a projective limit of topological vector spaces with the same property. For this to be true for Fréchet spaces, we would need the limit to be over a countable set of indices, and it does not seem to be the case. The proposition should be true as I guess we can restrict to a countable set of indices, and as the space is latter shown to be nuclear and thus reflexive if Fréchet. However, I was wondering if I was missing something about the stability of Fréchet and reflexive spaces. Are they stable by projective limits ?

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  • $\begingroup$ Of course, only countable limits of Fréchet spaces are Fréchet (for example, a product $\mathbb R^I$ is the projective limit of finite dimensional spaces $\mathbb R^E$ with $E\subseteq I$ finite, but $\mathbb R^I$ is metrizable if and only if $I$ is countable), but this seems to be the case in the proposition you mention. $\endgroup$
    – Jochen Wengenroth
    Commented Jan 2, 2023 at 16:50
  • $\begingroup$ ok, I might have misread, I supposed the m in the index was real but in fact it may not be. thanks ! $\endgroup$
    – Mar
    Commented Jan 3, 2023 at 10:17

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