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!



Poincaré lemma in infinite dimensions
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)](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf)


