In this paper, we introduce \(\ell ^p\) -information geometry, an infinite dimensional framework that shares key features with the geometry of the space of probability densities \( \textrm{Dens}(M) \) on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the \(\ell ^2\) -probability simplex with a noncanonical differentiable structure induced via the q-root transform from an open subset of the \(\ell ^q\) -sphere. This structure renders the q-root map an isometry, enabling the definition of Amari–Čencov \(\alpha \) -connections in this setting. We further construct gradient flows with respect to the \(\ell ^2\) –Fisher–Rao metric, which solve an infinite-dimensional linear optimization problem. These flows are intimately linked to an integrable Hamiltonian system via a momentum map arising from a Hamiltonian group action on the infinite-dimensional complex projective space.