TITLE: Poincaré lemma in infinite dimensions
QUESTION [6 upvotes]: 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!

REPLY [10 votes]: 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)