The real analysis portion of the test was one I really concentrated hard on, since it was also my biggest weakness. Here's what I've found for the most part:
- Know the basic properties of metric spaces, as well as examples and non-examples of metrics.
- You should know how to reason about the supremum and infimum of sets.
- For limits, rather than only thinking in terms of deltas and epsilons, know how to characterize limits in terms of sequences. It's a much easier way to deal with Dirichlet-like functions.
- Continuity of functions is a big real analysis theme. You need to be familiar with different kinds of continuity such as uniform and Lipschitz continuity, as well as what sorts of properties are preserved by continuity.
- Related to the above point, differentiability and integrability are also very important.
- Another common topic is sequences of functions. Understand the difference between pointwise and uniform convergence, and what kinds of properties are preserved by uniform convergence.
- In the many GRE tests I have taken, I have never once seen a question about Lebesgue measure or measurable functions. I believe it used to be tested on (based on the Descriptive Booklet at http://www.math.ucla.edu/~iacoley/gre/othertests/math97-99.pdf), but if it does end up coming up, don't expect more than a single question.
- Closely related to real analysis is topology. Questions about topology are sometimes asked in the context of . The Heine-Borel theorem comes up a lot when talking about compact sets.
The Princeton Review is definitely not
enough for real analysis, as it does not address different kinds of continuity or convergence, instead focusing on Lebesgue measure when it really doesn't need to. To prepare for real analysis, I'd instead strongly recommend A Problem Book in Real Analysis
by Asuman G. Aksoy and Mohamed A. Khamsi. Here's the review from my resource list (https://www.mathsub.com/resources/
Finding a good real analysis textbook for GRE review was one of the toughest parts of writing this list; many analysis books are drowning in notation and light on examples as you wade through proof after proof trying to find the nuggets of information you need to study. Aksoy and Khamsi’s book, on the other hand, is perfect for this purpose. Each section begins with a bulleted review of key definitions and theorems, and the problems that follow give immediate practice with the kinds of examples you want to have on quick recall for the GRE. The best part is that a full solution is given for every single problem at the end of each chapter!
Hope this helps!