Great day, got up at 8 & did my routine, practiced my italian for 30m then spent 3h reading chapter 2 of https://geometricalgebra.org then spent an hour reviewing what I learned. the rest of the day I spent writing up solutions to https://uli.rocks/abbott (3h), improved my analysis a good bit, though I still feel like I'm extremely slow at solving exercises.

Stopped at 6, spent 6-8 chatting on the math discord and eating dinner

Euler2 was talking about his proof of the Jacobi triple product using quantum mechanics again, I wonder if I could understand if I dedicated a few days to it and asked him questions? It's above my level but levels aren't real anyway

He also talked about q-analogs which look really cool, basically it's a number system that carries more information with it, in normal math if you multiply the sides of a rectangle to get an area you've lost the information of the side lengths, while with these numbers you'd be able to go backwards. They're used in combinatorics apparently, another field I need to practice

Watched an hour of youtube, then journaled

Stayed on my computer till 11 (bad!) I'm supposed to get off at 10, but I was chatting on discord.

Overall spent ~8.5h studying, if you count the productive part of the math discord chatting then ~9h.

Short summary of the analysis things I learned for the mathy people out there (mostly I just got better at proofs I think hope)

• 6.2.6 (b): If $(f_n)$ are bounded and $(f_n) \to f$ uniformly then $f$ is bounded (even if the $f_n$ are increasing bounded!)
• Examples of $(f_n) \to f$ pointwise where each $f_n$ has countably many discontinuities but $f$ has uncountably many.
• That we can use a triple-triangle inequality to show $|f(x)-f(y)| < 2\epsilon + |f_n(x)-f_m(x)|$ In other words, when $(f_n) \to f$ uniformly the behavior of $|f(x)-f(y)|$ is "basically" the same as $|f_n(x)-f_n(y)|$.
• That if $f$ is uniformly continuous then $f_n(x) = f(x+1/n)$ converges uniformly to $f$.