Real analysis

[thescienceboy]

Hey guys,

I think this topic came up here several times from Canadian applicants. It was said that "Real analysis" taken by US students can be different from the one offered at Canadian universities.
I have three courses at my university related to mathematical analysis.

Introduction to mathematical analysis:
Proofs in calculus and analysis. Topics include sets, functions, axioms for R, applications of the completeness axiom, countability, sequences and their limits, monotone sequences, limits of functions, continuity.

Principles of mathematical analysis:
Rigorous presentation, with proofs, of fundamental concepts of analysis: limits, continuity, differentiation, integration, fundamental theorem, power series, uniform convergence.

Real analysis:
Survey of the real and complex number systems, and inequalities. Metric space topology. The Riemann-Stieltjes integral. Some topics of advanced calculus, including more advanced theory of series and interchange of limit processes. Lebesgue measure and integration. Fourier series and Fourier integrals.

Which "Analysis" is the one mentioned all the time on this forum? And how do adcoms know what content is behind the titles of the courses?

[pevdoki1]

What people mean by "real analysis" is a combination of the latter two courses. It seems to me that any student taking the second course (real analysis) should know the material of the first (principles of mathematical analysis). In my school, these two courses form a year long sequence (Real Analysis I and II).

Take "Principles of mathematical analysis". It covers most of the fundamentals.

P.S. The " Introduction to mathematical analysis" is called "Intro to higher math" in my school. It's a prerequisite for the real analysis courses, but you don't really need it.

[israelecon]

i am not sure which one = real analysis in the US. but about the adcoms:
of the 3 courses you mention the first one sounds like "introduction to proofs", not bad, but not that impressive for an adcom. the second one "principles" is kind of ambiguous, but also if i were sitting on an adcom it would sound like an introductory course at best about calculus. btw, from the syllabus it sounds like thats exactly what it is.
real analysis is the only course that sounds like its on a very high level. of course, there are real analysis courses on a lower level, but i think this is where your schools reputation plays a role in signaling to the adcoms what kind of course this would be.

[pevdoki1]

I don't know, israelecon. I'm learning stuff in the third course now (Fourier Series, measure theory, Stieltjes integral etc), and it seems to me like this stuff wouldn't be very useful to economics. Who cares about upper and lower sums and pointwise convergence properties of Fourier series?

On the other hand, the "principles" course covers the basic things everyone should know how to use (Cauchy series, continuity, uniform convergence of functions etc). I would say the emphasis should be placed on that material, since it will probably come in useful. By the way, no one learns this stuff in U.S. calculus courses.

[israelecon]

true, pevdoki. but, i was assuming that adcoms are looking for level of math sophistication not level of useful math knowledge. this is debatable.

[Mr.Keen]

If you are a time series econometrician, measure theory is actually really useful. That being said, in my school "introduction to analysis" covered all the material in your "principles of analysis" class and part of the "real analysis" class. Measure theory was then covered in a course called real variables.

Also, pevdoki1, one of my recommenders, who is a Minnesota PhD told me: "what you really need is a course on measure" after I asked what math classes to take after senior level analysis. So maybe you'll find more use to those math skills of yours than you anticipate, .

[buckykatt]

My sense is that on TM, "real analysis" is mostly a shorter way of saying "an advanced course where the students are expected to produce original, nontrivial proofs on a regular basis".

So, for example, my school--like many US schools--has a course called Discrete Math (or Discrete Structures--I can never remember) that serves as the "intro to proofs" course. (Math and computer science majors typically take it their first year.) It can include a lot of different things, e.g. probability, number theory, graph theory, etc., but it's sure to include relations/functions, set theory, and plenty of practice writing proofs of all kinds. Personally, I think its main function is to weed out the Java code jockeys from the mathematicians, but I'd think that even this course would send a useful signal to an adcom that you aren't going to show up to math camp asking why the A's are upside down and the E's backward.

Our real analysis sequence is two semesters, and is required of all math majors. (Although we cover limits, power series, and such in the regular calculus sequence, the level of the students in those classes is very mixed, and the emphasis is on applications, so there's no expectation that students be able to reproduce the proofs that are presented in lectures.) As a rough guide, I'd say that it picks up somewhere in the middle of the first course you listed above and ends somewhere in the middle of the third. (We don't get into measure theory and the Fourier series gets no more than the occasional mention.) Many students at my school are education majors, so the focus is on the conceptual underpinnings of the calculus, but we managed to cover pretty much everything I know I'll need for the first year of a PhD program. I wish we had done more, but IMHO this course should satisfy any adcom that I'm ready to tackle PhD econ.

How do adcoms know what the content of a course is? Well, they probably don't, exactly. But I doubt that they care, exactly, so long as they're convinced that you really can read and write substantial proofs and know why you want your function defined on a compact set.

[Fatrapa]

I think Stieltjes integral is very useful, particularly in econometrics. I remembered seeing them many time and always being a bit confused by their property...

[Fatrapa]

And if you want to study Decision theory you will need an excellent knowledge of set theory...

[Internationalstudent08]

I finished with my real analysis take-home final a few days ago. I'm so happy this is over