I think that sometimes even more important than remembering statements by heart is remembering some key ideas in the proofs, because they become new tools in your toolbox. Another thing that helps is to try to make variations of the theorems in the book and try to prove or disprove them. Self study is hard but it is essentially the only way from the PhD and on wards. Also try to discuss with other mathematicians as much as you can.

@IosifPinelis, Ok, I see in the embedding of the circle in an infinite dimensional space one might loose uniqueness of the minimizer.

What about a circled centered at the origin in $\mathbb{R}^2$ ?

Thanks, I was not aware of this concept. Actually it is exactly the definition of weak sequentially completeness.

@AlekseiKulikov There seem to be 3 papers citing this one, but no one seems to contain a solution to this problem.