It was my last year in university, I had this Calculus book I got from the library open in front of me, while the professor emitted words that blurred into background noise. I glanced over at my glaring laptop and typed a few semi-coherent sentences and formulas, and repeatedly pressed the backspace. Each individual piece made sense, but I couldn't piece it together into a clear picture. Both my professor and the book kept hinting towards a bigger picture, but at the same time held a veil over my eyes, claiming it was out of scope, which only made me more frustrated. I had questions needed answering, and I wouldn't take "out of scope" as an answer.
I regressed more and more into my own writings, ignoring whatever the professor was trying to prove on the blackboard, kept going backwards and backwards to the very foundations of Calculus and Mathematical Analysis. I needed to build it from scratch. Topology seems like it may have some answers for me, and while I was writing down the definition of a topology and the properties of open sets, things finally started clicking. I still couldn't see the overall structure, but for the first time things kind of made sense.
So I dug deeper, and with each definition, with each theorem I got a faint glimpse into a world of pure beauty. It was like looking at an enormous mural, depicting this most complex pattern, that you can only see each small brush stroke at a time. But if you try and keep it in your mind, and you step back to absorb it all, this magnificent piece of art will take your breath away and leave you speechless and in awe. Like staring at the very essence of reality itself and saying: "ah, I see what you did there."
Both Set Theory and General Topology introduced me to an ensemble of interesting and recurring characters, each with their own character arc and a story to tell, each story thread neatly intertwined with the rest. I kept going and stumbled into Functional Analysis, curious where the story go from there.
What is the purpose of this sequence?
This sequence has no purpose unto itself, rather it is the product of my own studies and explorations. The only part with any purpose and a target audience is the short snippets of poorly written motivations I slapped on top each section to make it more accessible to anyone who might be interested to follow through.
Some stylistic choices
Each section should feel like it is aiming to reach a specific point, a grand theorem, or an important insight. I decided against using quantifier symbols, but rather use plain English to denote the universal quantifier ("for all") and existential quantifier ("exists"), and same for the logical operators "not", "and" and "or". I make heavy use of the abbreviations s.t. ("such that") and h.t. ("holds that"), and I use := for definitions.