  In this paper, we study the Seiberg-Witten equations on a compact 3-manifold
with boundary. Solutions to these equations are called monopoles. Under some
simple topological assumptions, we show that the solution space of all
monopoles is a Banach manifold in suitable function space topologies. We then
prove that the restriction of the space of monopoles to the boundary is a
submersion onto a Lagrangian submanifold of the space of connections and
spinors on the boundary. Both these spaces are infinite dimensional, even
modulo gauge, since no boundary conditions are specified for the Seiberg-Witten
equations on the 3-manifold. We study the analytic properties of these monopole
spaces with an eye towards developing a monopole Floer theory for 3-manifolds
with boundary, which we pursue in Part II.
