We introduce diffeological real or complex vector spaces. We define the fine diffeology on any vector space. We equip the vector space 𝓗 of square summable sequences with the fine diffeology. We show that the unit sphere 𝓢 of 𝓗, equipped with the subset diffeology, is an embedded diffeological submanifold modeled on 𝓗. We show that the projective space 𝓟, equipped with the quotient diffeology of 𝓢 by 𝓢¹, is also a diffeological manifold modeled on 𝓗. We define the Fubini-Study symplectic form on 𝓟. We compute the momentum map of the unitary group U(𝓗) on the sphere 𝓢 and on 𝓟. And we show that this momentum map identifies the projective space 𝓟 with a diffeological coadjoint orbit of the group U(𝓗), where U(𝓗) is equipped with the functional diffeology. We discuss some other properties of the symplectic structure of 𝓟. In particular, we show that the image of 𝓟 under the momentum map of the maximal torus 𝕋(𝓗) of U(𝓗) is a convex subset of the space of moments of 𝕋(𝓗), infinitely generated.