  In this paper we discuss how to define an appropriate notion of weak topology
in the Wasserstein space $(\mathcal{P}_2(H),W_2)$ of Borel probability measures
with finite quadratic moment on a separable Hilbert space $H$. We will show
that such a topology inherits many features of the usual weak topology in
Hilbert spaces, in particular the weak closedness of geodesically convex closed
sets and the Opial property characterizing weakly convergent sequences. We
apply this notion to the approximation of fixed points for a non-expansive map
in a weakly closed subset of $\mathcal{P}_2(H)$ and of minimizers of a lower
semicontinuous and geodesically convex functional
$\phi:\mathcal{P}_2(H)\to(-\infty,+\infty]$ attaining its minimum. In
particular, we will show that every solution to the Wasserstein gradient flow
of $\phi$ weakly converge to a minimizer of $\phi$ as the time goes to
$+\infty$. Similarly, if $\phi$ is also convex along generalized geodesics,
every sequence generated by the proximal point algorithm converges to a
minimizer of $\phi$ with respect to the weak topology of $\mathcal{P}_2(H)$.
