I'm about to start a PhD in an optimization lab, and I'd like to study some analysis beforehand.
So far, I've studied linear, non-linear, and integer optimization during my undergrad courses, and they were pretty theoretical. However, I've heard that functional analysis and analysis on manyfolds also hold many important results for optimization (mostly for non-linear), and I'd like to take a look before the PhD starts. For functional analysis, my uni uses the Kreyszig's book, and a random one for analysis on manyfolds.
However, I'm aware that only a few chapters in these books are directly relevant to optimization. For instance, in Kreyszig's book, I know that the sections on metric spaces and Banach's fixed point theorem are essential, but I'm uncertain about the rest.
In a nutshell: Could you guys recommend me some books and chapters in each book that will be most helpful for this purpose? :)