Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$.
The most remarkable result of these topologies is the Banach-Alaoglu-Bourbaki, which asserts that for the weak$^*$ topology (Over $E^*$), the unit ball: $$B_{E^*}=\{f \in E^* | \ ||f|| \leq 1 \}$$ is compact. We know that in infinite dimensional Banach spaces the unit ball is not compact over the strong topology (the topology generated by $||\cdot||$).
The last part of the book introduces Sobolev Spaces and the Variational Method for solving PDEs, applying Functional Analysis (Particularly Hilbert Spaces). But it never really uses weak topologies. He does use them to show the existence of the projection in Hilbert Spaces (which by the way can be done more simply), but aside from that it is never "applied".
I can understand the importance of the Banach-Alaoglu-Bourbaki theorem, as a metric topology where the unit ball is not compact is not very suitable, and the book gives certain results (Corolary $3.23$, for example, which asserts that a lower semicontinuous convex function achieves its minimum; or Milman-Pettis Theorem) which show its usefulness in a general context.
But provided this book aims to apply the theory provided to PDEs,
I wonder if there are examples of PDEs for which Weak Topologies aid when applying, for example, the Variational Method.
