Abstract: 
\def\TT{{\sym T}}\def\UU{{\sym U}}\def\cH{{\scr H}}\def\cS{{\scr S}}\def\cP{{\scr P}}We introduce diffeological real or complex vector
spaces. We define the fine diffeology on any vector space.
We equip the vector space $\cH$ of square summable
sequences with the fine diffeology. We show that the unit
sphere $\cS$ of $\cH$, equipped with the subset diffeology,
is an embedded diffeological submanifold modeled on $\cH$.
We show that the projective space $\cP$, equipped
with the quotient diffeology of $\cS$ by $\cS^1$, is also
a diffeological manifold modeled on $\cH$. We define the
Fubini-Study symplectic form on $\cP$. We compute the
momentum map of the unitary group $\UU(\cH)$ on the sphere $\cS$
and on $\cP$. And we show that this momentum map identifies
the projective space $\cP$ with a diffeological coadjoint
orbit of the group $\UU(\cH)$, where $\UU(\cH)$ is equipped with the
functional diffeology. We discuss some other properties of
the symplectic structure of $\cP$. In particular, we show
that the image of $\cP$ under the momentum map of the maximal
torus $\TT(\cH)$ of $\UU(\cH)$ is a convex subset of the
space of moments of $\TT(\cH)$, infinitely generated.

1. Einstein Institute
   The Hebrew University of Jerusalem
   Campus Givat Ram
   Jerusalem 91904, Israel