  In this paper we use a diffeo-geometric framework based on manifolds that are
locally modeled on "convenient" vector spaces to study the geometry of some
infinite dimensional spaces. Given a finite dimensional symplectic manifold
$(M,\omega)$, we construct a weak symplectic structure on each leaf ${\textbf
I}_{w}$ of a foliation of the space of compact oriented isotropic submanifolds
in $M$ equipped with top degree forms of total measure 1. These forms are
called weightings and such manifolds are said to be weighted. We show that this
symplectic structure on the particular leaves consisting of weighted Lagrangian
submanifolds is equivalent to a heuristic weak symplectic structure of
Weinstein [Adv. Math. 82 (1990), 133-159]. When the weightings are positive,
these symplectic spaces are symplectomorphic to reductions of a weak symplectic
structure of Donaldson [Asian J. Math. 3 (1999), 1-15] on the space of
embeddings of a fixed compact oriented manifold into $M$. When $M$ is compact,
by generalizing a moment map of Weinstein we construct a symplectomorphism of
each leaf ${\textbf I}_{w}$ consisting of positive weighted isotropic
submanifolds onto a coadjoint orbit of the group of Hamiltonian
symplectomorphisms of $M$ equipped with the Kirillov-Kostant-Souriau symplectic
structure. After defining notions of Poisson algebras and Poisson manifolds, we
prove that each space ${\textbf I}_{w}$ can also be identified with a
symplectic leaf of a Poisson structure. Finally, we discuss a kinematic
description of spaces of weighted submanifolds.

Differential Geometry;Symplectic Geometry