Timeline for Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

6 events

when toggle format	what		by	license	comment
May 14 at 23:49	vote	accept	David Roberts♦
May 14 at 11:52	comment	added	Tobias Diez		I like this argument! (For completeness: it is enough to show that the subspaces are closed since the Banach-Schauder theorem applies in the Frechet setting and shows that the decomposition is then a topological isomorphism.)
May 14 at 11:43	comment	added	David Roberts♦		It is the fact that $d$ and $\delta$ have closed images, which is non-trivial. <--- yes, this was the part that seemed the most mysterious!
May 14 at 11:19	comment	added	Will Sawin		The integration over submanifolds trick is also helpful for understanding variation as the metric varies since it shows that the image of $d$ is fixed as the metric varies and the image of $\delta$ is the translate of a fixed closed subspace by a smoothly varying isomorphism of vector bundles, so both are closed in families.
May 14 at 11:18	comment	added	Branimir Ćaćić		The argument from Poincaré duality is very nice indeed!
May 14 at 9:20	history	answered	Stefan Waldmann	CC BY-SA 4.0