The infinite Hopf fibration
March 29, 2006, Update October 5, 2006
In this paper, I introduce diffeological vector spaces, and the fine diffeology. I illustrate these definitions on the example of the infinite Hopf fibration. The infinite Hopf fibration is the projection of the unit sphere of the Hilbert space of complex summable series, to the infinite projective space, its quotient by the action of the circle. The infinite projective space is equipped with a generalization of the Fubiny-Study 2-form. I show how the infinite projective space equipped with this 2-form is a coadjoint orbit of the unitary group of the Hilbert space, equipped with the functional diffeology.

Text delivered at the 7th Conference on Geometry and Topology of Manifolds, The Mathematical Legacy of Charles Ehresmann, May 8-15, 2005 Bedlewo, Poland . 
Published in Geometry and Topology of Manifolds, Banach Center Publications, volume 76, Institute of Mathematics, Polish Academy of Sciences, Warszawa 2007.