MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
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)

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.