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What is Real Analysis?

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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.
A slightly more complex description is given by Steve Zelditch at Johns Hopkins University:
    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.
As you can see, real analysis is a somewhat theoretical field that is closely related to mathematical concepts used in most branches of economics such as calculus and probability theory.

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
There are two key reasons why those entering a graduate program in economics should have a strong background in real analysis:
  1. Topics covered in real analysis, such as differential equations and probability theory are used extensively in economics.

  2. Graduate students in economics will commonly be asked to write and understand mathematical proofs, skills which are taught in real analysis courses.
Prof. Olmstead saw practicing proofs as one of the core objectives of any real analysis course:
    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.
Thus if a real analysis course is not available at your college or university, I would recommend taking a course in how to write mathematical proofs, which the mathematics departments of most schools offer.

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

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