**Question:**What is Real Analysis?

**Answer:**

**[Q:]**I read your article Books to Study Before Going to Graduate School in Economics and saw that you mentioned something called “real analysis”. What do you learn in a real analysis course? What do you need to know before you take a real analysis course? Why is taking a real analysis course helpful if you’re planning to do graduate work in economics.

**[A:]**Thanks for your great questions.

We can get a feel for what is taught in a real analysis course by taking a look at a couple of real analysis course descriptions. Here’s one from Margie Hall at Stetson University:

- Real analysis is a large field of mathematics based on the properties of the real numbers and the ideas of sets, functions, and limits. It is the theory behind calculus, differential equations, and probability, and it is more. A study of real analysis allows for an appreciation of the many interconnections with other mathematical areas.

- Real Analysis is an enormous field with applications to many areas of mathematics. Roughly speaking, it has applications to any setting where one integrates functions, ranging from harmonic analysis on Euclidean space to partial differential equations on manifolds, from representation theory to number theory, from probability theory to integral geometry, from ergodic theory to quantum mechanics.

To be comfortable in a real analysis course, you should have a good background in calculus first. In the book *Intermediate Analysis* John M.H. Olmstead recommends taking real analysis fairly early in one’s academic career:

- ...a student of mathematics should properly begin to make his acquaintance with the tools of analysis as soon as possible after the completion of the first course in calculus

- Topics covered in real analysis, such as differential equations and probability theory are used extensively in economics.
- Graduate students in economics will commonly be asked to write and understand mathematical proofs, skills which are taught in real analysis courses.

- In particular the student should be encouraged to prove (in full detail) statements which previously he has been persuaded to accept because of their immediately obviouness.

I wish you the best of luck in your preparations for graduate school!

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