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Visible Hand
10-17-2008, 11:51 AM
Hi folks,

I am an international student in the last year of a Master Program and I'm applying this fall to Econ PhDs.

My undergraduate/graduate background is not that excellent but neither so poor: I took uni/multivariate calculus, linear algebra, differential equations, mathematical statistics, multivariate statistics, basic topology, static and dynamic optimization, bayesian inference theory.

My graduate advisor suggested me to improve my math background in order to increase my admission chances; the point is that I need to rigorously study Real Analysis, at least at the undergraduate level.

Therefore as part of my "free credits" I chose to take a course in the Math/Physics school, in mathematical analysis. This course deals with functional analysis in R^n, some measure theory, Lebesgue integration and further advanced stuff with many applications in Physics.

With the support of my graduate advisor I spoke with the Math Professor of this course in order to set a personalized program for me and another girl coming from the MS in Economics (and willing to applying to PhDs) - since these are "free credits" we are allowed to do this. The Math Professor advice was this: to stop attending the course after Lebesgue integration and to make a personalized oral examination over about HALF of the course programme PLUS an additional part from RUDIN's book according to our personal needs in Real Analysis. I and this girl will study alone this additional part, however the Prof. is available to explain us what is not very clear in the office hours.

Since the work I am doing I think will show up in some LoR (perhaps the one of my graduate advisor who is in touch with the Math Prof.) I want to do it well. The point is that I basically have to choose the topics I have to study alone. The math Prof. suggested them to be L^p spaces and Fourier series. However I do not believe these are the most useful for an Economics-oriented student. What do you guys suggest?!

The point is that I have to discuss with this prof. primarily the book I have to study on. Which one do you suggest?

Principles of Mathematical Analysis by W. Rudin
Principles of Mathematical Analysis by Walter Rudin - Math Books at Apronus.com (http://www.apronus.com/math/rudinreal.htm)

which is simpler but still covers rigorously many topics I always did not very formally (i.e. set theory, sequences etc.) OR

Real and Complex Analysis by W. Rudin
Real and Complex Analysis by Walter Rudin - Math Books at Apronus.com (http://www.apronus.com/math/rudincomplex.htm)

I understand doing the second would be better, but I don't want it to be too hard, especially when it is better for me to make more rigorously some previous stuff.

My question on the two books is: on which of them did you guys (you who studied undergraduate Real Analysis) study on?! Which of them corresponds more to a "standard" undergraduate course in Real Analysis?!

Thank you in advance for any provided feedback.

jeeves0923
10-17-2008, 12:11 PM
Many topics in Rudin are more or less assumed that you know before measure theory and Lebesgue integration... so it might be weird doing them in the opposite order.

From Rudin, things with obvious implications to economics are:
Point Set Topology, Sequences and Series, Continuity and Differentiation, and possibly Sequences and Series of Functions. Riemann integration is good to know, but once you have Lebesgue, life gets much simpler...

unitroot
10-17-2008, 01:43 PM
All I can say about the listed texts is that "Real and Complex Analysis" is a graduate level (usually taught at the first year PhD-level) math text. This and other similar texts are generally used only in courses where it is assumed you have already taken a course based on "Principles of Mathematical Analysis" or a similar book.

I don't know whether you will find any use for these materials actually useful in your work in economics. For example, people who work in empirical/applied economics and/or applied econometrics rarely if ever need any real analysis for their work. Advanced real analysis might be useful in advanced theory or in some kinds of econometric research.

The applications of Fourier analysis are probably limited in economics but it can be helpful in studying the spectral decompositions of time series (which are often done with Fourier transforms). L^p spaces and functional analysis are generally very cool to know if you want to study theory. Theoretical asset pricing, general equilibrium, game theory, and many be other subjects in theory make use of it.

jeeves0923
10-17-2008, 01:49 PM
I don't know whether you will find any use for these materials actually useful in your work in economics. For example, people who work in empirical/applied economics and/or applied econometrics rarely if ever need any real analysis for their work. Advanced real analysis might be useful in advanced theory or in some kinds of econometric research.



I'm taking PhD Metrics now, and I'm very glad to have had Real Analysis. Very useful. I'm not sure if all PhD Metrics courses are as rigorous as the one I'm doing now, but there are a lot of proofs, and certainly a lot of concepts that come a lot quicker when you know basic topology and sequences.

YoungEconomist
10-17-2008, 02:09 PM
There's a textbook I was asking about recently titled "Real Analysis with Economic Applications," and if you search for my recent post (or even just google the title) you are bound to find it. The author claims that this textbook covers the topics that are most critical to econ students, so you might want to take a look at the table of contents before you finalize your list of topics.

Visible Hand
10-17-2008, 04:57 PM
Thank you guys all.

My doubt remains the same: perhaps I need to revise some stuff from the first book (IMPORTANT: where you seemingly find the sequences/topology stuff you say to be important in micro/metrics graduate courses), which seems too simple anyhow, but on the other hand the second Rudin, being a graduate textbook aimed at Math PhD students, is certainly not what I need!


There's a textbook I was asking about recently titled "Real Analysis with Economic Applications," and if you search for my recent post (or even just google the title) you are bound to find it. The author claims that this textbook covers the topics that are most critical to econ students, so you might want to take a look at the table of contents before you finalize your list of topics.

I know it: from Ok Table of Contents for Ok, E.: Real Analysis with Economic Applications. (http://press.princeton.edu/TOCs/c8274.html) .

Perhaps the best for me is to revise the first 100 pages (you can download them) and to study the next chapters up to page 250-300 ca.

Keep also in mind that I want to become an applied economist, certainly not a theorist, and that I want this course to maximize my Apps-chances (that is, my Ref. writers think this course is effective to an economist and corresponds to what you guys typically in the US study as undergraduate Real Analysis), not my math-knowledge.

Thank you again. BTW, does anyone of you have a syllabus from some previous undergrad RA course?!

pookie bear
10-17-2008, 05:11 PM
check MIT open courseware...

Visible Hand
10-17-2008, 05:42 PM
check MIT open courseware...

Hum I can only find a syllabus for "Analysis I" which corresponds to what I was used to call "calculus", at a level below Rudin's principles maybe. If it is so, I have a stronger math background than I was used to think. I have been thinking that Real Analysis has to do with extensive application of Topology and extentions of Calculus to metric spaces different than those characterized by the Euclidean metric.

pookie bear
10-17-2008, 08:44 PM
There are several sections of Analysis I up there. I believe the older ones use Rudin. I don't believe they have lecture notes up for Analysis, but if you search the web and even econphd you will find something relevant.