Starting on January 5th, the beginning of the winter quarter here at UCLA, I will be teaching Math 245B, a graduate course on real analysis. As the name suggests, the course is a continuation of the Math 245A course that just concluded in this fall quarter, taught by Jim Ralston, who covered the basics of measure theory and spaces. In this quarter, I plan to cover more of the foundational theory of graduate real analysis, specifically
- Signed measures, Radon measures and the Radon-Nikodym theorem;
- The general theory of spaces;
- Introduction to functional analysis, particularly the theory of Hilbert spaces and Banach spaces;
- Various aspects of point set topology of relevance to analysis, including Tychonoff’s theorem and the Stone-Weierstrass theorem.
I will be using Folland’s “Real analysis” as a primary text and Stein-Shakarachi’s “Real analysis” as a secondary text. These two texts already do quite a good job of covering the above material, but it is likely that I will supplement them as the course progresses with my own lecture notes, which I will post here, though I do not intend to make these notes nearly as self-contained and structurally interlinked as my notes on ergodic theory or on the Poincaré conjecture, being supporting material for the main texts rather than a substitute for them.