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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?


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1 Answer 1

up vote 10 down vote accepted

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

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