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In this paper we discuss how to define an appropriate notion of weak topologyin the Wasserstein space $(\mathcal{P}_2(H),W_2)$ of Borel probability measureswith finite quadratic moment on a separable Hilbert space $H$. We will showthat such a topology inherits many features of the usual weak topology inHilbert spaces, in particular the weak closedness of geodesically convex closedsets and the Opial property characterizing weakly convergent sequences. Weapply this notion to the approximation of fixed points for a non-expansive mapin a weakly closed subset of $\mathcal{P}_2(H)$ and of minimizers of a lowersemicontinuous and geodesically convex functional$\phi:\mathcal{P}_2(H)\to(-\infty,+\infty]$ attaining its minimum. Inparticular, we will show that every solution to the Wasserstein gradient flowof $\phi$ 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 aminimizer of $\phi$ with respect to the weak topology of $\mathcal{P}_2(H)$.

Weak topology and Opial property in Wasserstein spaces, with  applications to Gradient Flows and Proximal Point Algorithms of geodesically  convex functionals