  In this series of papers, we investigate the projective framework initiated
by Jerzy Kijowski and Andrzej Oko{\l}\'ow, which describes the states of a
quantum (field) theory as projective families of density matrices. The present
first paper aims at clarifying the classical structures that underlies this
formalism, namely projective limits of symplectic manifolds. In particular,
this allows us to discuss accurately the issues hindering an easy
implementation of the dynamics in this context, and to formulate a strategy for
overcoming them.
