This series of lectures consist of three parts and is a preparation of prelims for MSRI summer school “Recent Topics in Well Posedness” which will take place in the July of 2022 at NCTS. For the information regarding the MSRI summer school, see

https://www.msri.org/summer_schools/964

One motivation for introducing the weak convergence comes from the calculus of variation. For example, the minimizer of certain energy functional solves the corresponding Euler-Lagrange equation. However, the usual technique, known as the compactness argument, for finding the extreme value(including minimum) does not hold true in the infinite-dimensional setting with the usual strong topology. As a result, the notion of weak convergence comes into rescue and we have the so-called weak* compactness, known as the Banach-Alaoglu theorem.

To begin with, after discussing the motivation, I will introduce the weak convergence from a topological viewpoint. In the second place, I will introduce the weak* convergence and prove the famous Banach-Alaoglu theorem. Finally, I will prove the weak compactness of the closed unit ball in the reflexive Banach space and use this result to find the minimizer of a certain class of energy functional as an application.