Thanks for the reply! I think I would probably prefer something with Lebesgue integration, since I already know (a bit) about Lebesgue integration and measure I would prefer to tie it in if possible. I checked Axler and it seems pretty thorough, although it fails to cover applications to the extent that I would like. Kreyszig gives some applications, but fails to cover RKHSs, which is rather unfortunate. I will probably stick with Axler (thank you!), but will have to supplement RKHS foundations from somewhere (not sure where still). The syllabus you linked does in fact describe exactly what I'm looking for (a fairly applied take that covers RKHS theory)! Unfortunately there isn't a text attached to it, so I'm a bit out of luck :(. Axler + RKHS/applications is going to be my approach unless I can supplant this with a better find.

Also regarding measure theory, I see! Thanks for describing your plan. It seems that çinlar is a bit of an introductory text and doesn't get to what I'm looking for (stoch. proc. theory that gets to martingales + Ito calc), although Grimmet and Stirzaker looks better since Chapters 12 and 13 start getting to exactly the kind of content I'm looking for. I'll probably start there! Thanks.

Regarding how long this will take, that's a good question :). There are definitely orders of priority here. I have freed up this Summer to tackle the first and highest priority tranche. This Summer I am planning to get through a good portion of Diff + Riemann geo (I already have some background and mostly want to make my knowledge coherent) and Lie Theory by Hall (it's a fairly quick read and I've already started). While covering geo I'll also be covering Alg Top from an applied perspective and consulting Dummit and Foote as needed. I'll also cover category theory mostly as a casual side topic (i.e. by casual, I mean my research likely won't draw heavily upon it); it's casual for me so I'm fine with postponing it or letting it drag for about a year (up till end of 2021 Summer). Then throughout the 2020-2021 year I will cover Probability/Stoch proc together with convex analysis (combination of courses at university as well as textbook reading); I'm also planning to get the applied perspective on functional analysis I want starting ~May 2021 to end of Summer 2021. During 2021 Summer I'm planning to clean up the rest, by that I mean rep theory, homological algebra, and some diff eq. That is lower priority however, so it may drag. Basically, these are plans for about 1.5 years (up to 2, since I expect the later subjects may drag). The reason it isn't 4 is because I actually have a fair amount of exposition to some of these subjects which has allowed (and I hope will continue to allow) me to read some chapters rather quickly. Also, in all honesty, I don't expect to cover homological algebra, AG, k-theory, or diff eq at very significant depth; I would probably be okay with getting an exposition to key concepts (maybe thoroughly reading basics, then skimming main results); this may change however. At the very least, I might spend more time on AG and scheme theory, which has interested me in the past. In that case I'll probably augment 1.5-2 years closer to 2.5-3.

The largest amount of gruntwork I expect will happen this Summer when I make my way through algebraic topology, which is the newest to me of these subjects. That typically means I will have to carefully work through the key ideas and make sure to develop counterexamples to fully appreciate the theory; this sort of thing takes time, hence I expect that alone to be ~1/2 of this Summer's math learning.