I will premit that I'm doing a PhD in partial differential equations (from a functional analysis perspective). I got a degree in mathematical engineering 2 years ago (so not exactly mathematics, in fact I lack of a complete preparation in General Algebra (I only know linear algebra), advanced geometry (differential geometry) and I know very little about topology). However in the meantime I do research in PDE's I wanted to study and to put togheder all the knowledge that I acquired at the university.
To this purpose I wanted to write some personal notes (in Latex) that start from the very bottom of the things I learned and go to PDE's. In particular I like to go from the very general things to the particular, not the other way aroung (which I consider great for the initial understanding but then very limiting and more time consuming once you have that initial understanding) I thought of something like this:
- Naive set theory (this will be enough for me)
- Point-set Topology (starting from topological space and going throgh all the important topological concepts, e.g. limit points, compactness, continuous functions, that if I looked at them only on $\mathbb R$ or a metric space in general it would be a very limited perspective). In this part also the concept of metric space and metrization would be investigated. For this study I would refer to the introductory book [SIDNEY A. MORRIS, TOPOLOGY WITHOUT TEARS1]
- Abstract Measure Theory. In this way I would introduce the concepts of measures and of Lebesgue integrals. I would probably refer to my notes at university.
- Functional analysis on infinite dimensional vectors spaces (so I will miss analysis on metric spaces, but that's ok for my purposes). Here I would start from Vector spaces and I would go to normed space (leaving out seminorms, local convexity for which I would only refer to books), Banach spaces and Hilbert spaces. Unbounded operators and adjoints would be introduced. In this part Distributions and Sobolev spaces are introduced. I would refer to [Peter Lax, Functional analysis].
- Semigroups of linear operators and applications to PDE's, reaching my goal.
In all of this the things that I give for granted are basic calculus on Real (or complex) spaces, so that when introducing the abstract concepts I have in mind that these are some generalizations of the ones in $\mathbb R^n$.
I plan to write a first "quick" edition in 1 year where most proofs are omitted for the sake of time (but I will refer to precise theorems to specialized books), because this notes will permit me to organize the ideas in my mind with a bottom up structure, so at first I'm not interested in writing down proofs. Then in future I will probably complete the missing proofs (and wriring them in a way that I really understand them).
What do you think? Am I missing something or do you have any suggestion? Or do you know someone that wrote something similar, that I can find online, so that I can take inspiration from that?
I'd definitely recommend taking a good, rigorous course on each of these low-level ideas at some point (or maybe you already have for some/all of them). This will automatically result in creation of some amount of notes, old homework assignments, etc which you could reference later if needed. It will also just force you to practice with the relevant concepts. I think it's important to have a "$\ge$ reasonably solid" understanding of the foundations of your area.
However, your suggestion here sounds closer to writing shortish textbooks on each of these low-level areas. I think this is much more investment than required to do your PhD work well. In my PhD program I did see a few people write out very extensive documents like this, but I would only recommend it as a fun/interesting side project. Which is not a bad thing - successful math students are usually the ones who just really enjoy math and often find themselves studying/learning extra math stuff on the side! But you should understand that this project is not your main priority, and don't let it get in the way of your main tasks.
Source: finished my math PhD at UCLA in 2017; spent a bunch of time around math grad students both before and after that.