# Poincaré lemma in infinite dimensions

Hi everyone,

Is the Poincaré lemma true in infinite dimensions?

Here's a precise statement:

Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true that every closed (smooth) $1$-form on $U$ is exact?

Thanks!

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## 1 Answer

Yes, it is, on convenient locally convex vector spaces. Convenient is a very weak completeness condition. See 33.20 in:

Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997.(pdf)

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Cool! Thank you. –  seub Nov 22 '12 at 2:20
And your book looks fantastic, thank you for that. –  seub Nov 22 '12 at 2:28