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Abstract

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