20204_math_580

This took place during my “LaTeX transition phase.” Here you will notice many visual components similar to what I am using currently (2022), but some from fall 2022 also remain. I was experiencing with new packages, e.g., tcolorbox, and was not doing very well with handling page breaks :)

Obviously, USC does not have a winter semester. My idea of learning functional analysis was all but a whimsical one — it started during my last meeting with Prof. Ożański for fall 2020’s MATH 490. Knowing I finished MATH 425a, he asked if I wanted something a little challenging. Turned out he taught this MATH 580 in fall and had recordings saved. He said his 580 didn’t rely on measure theory (525a) or really anything too advanced; instead, he focused the course on a different branch of analysis, and my bare background of 425a was enough.

Don’t worry about the measure theory, it is perhaps best to just take a course on it. I think Math 425b does it. It is a lot of fun by the way, but definitely worth studying in detail. For now just treat these problems with $L^p$ spaces as if it was Riemann integral, and only keep in mind that there are null sets. Namely, we identify any two functions that differ on at most a null set (it will be only relevant in PS2.4a). The course doesn’t rely on measure theory (almost) at all. I only use the $L^p$ spaces to give some examples, but they are mostly about the geometry of the $L^p$ norms (such as the Holder inequality), and not about the details from measure theory.

So on I went.

MATH 580, Functional Analysis (not registered), supervised by Prof. Wojciech Ożański.

• Lecture notes
• Homeworks (these were reviewed by Prof. Ożański so they at least won’t be ridiculously wrong) 1 (linear operators, p-norms, and L^p inclusions) ‖ 2 (Banach spaces & different proof of Uniform Boundedness Principle) ‖ 3 (inequalities and identities) ‖ 4 (Hilbert spaces & spectra of operators) ‖ 5 (more on operators)) ‖ 6 (reflexive spaces & weak convergence)