  We introduce a notion of a weak Poisson structure on a manifold $M$ modeled
on a locally convex space. This is done by specifying a Poisson bracket on a
subalgebra $\cA \subeq C^\infty(M)$ which has to satisfy a non-degeneracy
condition (the differentials of elements of $\cA$ separate tangent vectors) and
we postulate the existence of smooth Hamiltonian vector fields. Motivated by
applications to Hamiltonian actions, we focus on affine Poisson spaces which
include in particular the linear and affine Poisson structures on duals of
locally convex Lie algebras. As an interesting byproduct of our approach, we
can associate to an invariant symmetric bilinear form $\kappa$ on a Lie algebra
$\g$ and a $\kappa$-skew-symmetric derivation $D$ a weak affine Poisson
structure on $\g$ itself. This leads naturally to a concept of a Hamiltonian
$G$-action on a weak Poisson manifold with a $\g$-valued momentum map and hence
to a generalization of quasi-hamiltonian group actions.
