Real Analysis 1: What Topics Should Be Covered?

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Suffusion of Yellow
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Real Analysis 1: What Topics Should Be Covered?

Postby Suffusion of Yellow » Thu Jan 14, 2010 11:17 pm UTC

In your opinion. In my school they did series and sequences, epsilon delta proofs for derivatives and continuity, & the Darboux integral, but no mention of metric spaces, topology, so forth, which I seem to find in a lot of other books on Analysis 1. I'm guessing there's a debate over what should be taught?

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Re: Real Analysis 1: What Topics Should Be Covered?

Postby t0rajir0u » Thu Jan 14, 2010 11:43 pm UTC

I think most people more or less cover what's in baby Rudin. At MIT, for example, Analysis I covers up to Chapter 8. It sounds like your course may simply be less ambitious.

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Re: Real Analysis 1: What Topics Should Be Covered?

Postby MidsizeBlowfish » Fri Jan 15, 2010 12:27 am UTC

I think it depends on what kind of department that you take the course in. My undergrad school had a very applied math dept--what you said sounds like exactly what we covered in analysis I. If you wanted go to graduate school, it was strongly suggested you also take analysis II and learn about the more abstract things. I think it was a decent way to learn it...you essentially worked through a special case for an entire semester, so the generalizations in analysis II seemed somewhat natural. I can understand people who prefer to go straight through those things in one course though.

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Re: Real Analysis 1: What Topics Should Be Covered?

Postby skeptical scientist » Fri Jan 15, 2010 12:51 am UTC

It seems very strange to do analysis without metric spaces or topology, since so many of the theorems in analysis apply in much more general settings than just the real numbers with basically the same proofs, but you lose all of this generality if you are constantly having to use the least upper bound property instead of completeness and/or compactness. Also, these notions are not that hard, and are extremely useful.
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Re: Real Analysis 1: What Topics Should Be Covered?

Postby stephentyrone » Fri Jan 15, 2010 12:55 am UTC

My real analysis course did point set topology, metric spaces, sequences and series, continuity, and differentiation in the first semester (I may be forgetting one or two topics). Personally, I think the exact material isn't terribly important, what's important is learning to think and do rigorous proofs the way an analyst does them. Similarly, a first course in algebra (for me) is really about learning to think like an algebraist, not groups, rings and fields.
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Re: Real Analysis 1: What Topics Should Be Covered?

Postby romulox » Fri Jan 15, 2010 6:36 am UTC

You may need to consider how the course will be taught. My first real analysis course covered sets, sequences, limits, functions, continuity and differentiation from scratch using the Moore method. The first day of class our professor handed out a sheet of theorems. The class would not get a new sheet until the previous was completed. The second course developed the Riemann integral and some useful tools, also from scratch.

Along the way, quite a few topological (more or less restricted to the usual topology on R) theorems found their way in. This offers the benefit of students "figuring out"* and proving theorems like the heine-borel thm or the bolzano-weierstrass thm themselves, without the aid of a book, tutor, teacher, student or otherwise. It is also helpful to have students playing with topology before their first topology course.

Another consideration is the scope of various topics. For the most part, my two real analysis courses were restricted to interesting topics in R. Differentiation in R^n had a course mostly devoted to it, as did metric spaces and point-set topology respectively.

One of the shortcomings of the method was that few counterexamples were covered, mostly due to time constraints. Also this method is not for the faint of heart (student or professor) as strong proof writing is a necessity as is an attentive instructor. The class should also be mostly comprised of similarly ignorant (but intelligent and willing) students, as too far one way or the other can throw off the pace of the class. (I.e. start a seperate/honors section for the brilliant)

*"Figures out" is somewhat of a misnomer as the professor or instructor guides the student along some path with the previous theorems they have selected. A good deal of the success of the method comes from the instructor selecting a challenging but manageable path, that provides insight without giving away the results.

Hope this helps.

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gorcee
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Re: Real Analysis 1: What Topics Should Be Covered?

Postby gorcee » Fri Jan 15, 2010 4:27 pm UTC

It's also worth noting that there's a generational thing going on in collegiate education, at least in the US.

Cool story bro time:

I took Calculus I in my senior year of high school (1999-2000) and we learned differentiation formally through epsilon-delta limits. When I went to college, I was in the middle of some experimental curriculum planning, and calculus was taught with the aid of computers. We still had to learn calculus techniques, but we also had a heavy focus on the computer.

Before I graduated, I got sick and took 3.5 years off. When I returned to school to finish my last semester, the entire math curriculum had changed. The Maple-based calculus courses disappeared, but the courses were also not being taught with any formalism. Instead, they were focused on applications. Bear in mind, that although I went to an engineering school, the calculus courses were the same for all majors, so the emphasis was dedicated more towards developing the tools to recognize problems and solve them, rather than to develop a rigorous basis in mathematical formalism.

When I took Real Analysis after returning to college, the professor asked if anyone had seen epsilon-delta limits. I was the only one to raise my hand.

In short, what a Real Analysis class should cover is largely dependent on the prior education of the students. It's not bad to have to spend more time on those things, but if the early calculus courses are going to skip much of the formalism, then the curriculum needs to be adjusted so that undergrads can take a sequence of Analysis I/II and have a sufficiently strong rigorous background in math to take into grad school, if they choose. Under no circumstance, IMO, should a student leave Real Analysis I having never learned formal definitions of integration and differentiation, even if that comes at the price of losing out on some topology.

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Sygnon
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Re: Real Analysis 1: What Topics Should Be Covered?

Postby Sygnon » Sun Jan 17, 2010 11:57 am UTC

at my undergrad we did rudin, every problem up to and including chapter 9.

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Re: Real Analysis 1: What Topics Should Be Covered?

Postby Yakk » Sun Jan 17, 2010 3:33 pm UTC

What kind of material are you expected to have done before Real I? At my undergrad school, there where multiple different Real I courses, that had multiple different expected feed-in options.

The two I remember where the Applied Math and Pure Math versions of Real I.

Pure Math real I expected most of the people in it to have taken the advanced/theoretical branch of Calculus I-3, Classical Algebra, and Linear 1 and 2 in their first year, so they presumed everyone had already taken a bunch of proof-centric analysis courses already. It was mostly a course on topology and analysis in infinite dimensional vector spaces, if I remember rightly. I found it was somewhat funny that the "real analysis" course was the course where we threw away the real line and rebuilt much of what we had build on the real line in previous courses.

The applied math branch didn't presume that (although many had), so was more attached to the real line.

Real 2 (in the pure math branch) ended up being introduction to Functional and Fourier series, IIRC.

I do remember using Baby Rudin (I love that baby rudin hits that book on I'm feeling lucky) for some material in some course (I still have a copy of it). I'm not sure what course we used it in, but we didn't teach from the text -- the text was merely there as an additional reference in case we had problems getting how the professor was teaching the material, and occasionally as a source of problems (when the prof got lazy, I suspect).

So, throwing the question many of us asked back at the OP: what kind of prep courses are expected/required (note that these are different: it was expected that people had taken the theoretical first-year courses, but not required)? What kind of program/institution is this (an engineering school, a liberal arts school, or a course only for math majors, would all have different kinds of real 1 courses).
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z4lis
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Re: Real Analysis 1: What Topics Should Be Covered?

Postby z4lis » Sun Jan 17, 2010 4:54 pm UTC

My professor spent the first semester covering 1-dimensional calculus. It was assumed that none of us had seen derivatives or integrals put forth rigorously (which was correct for me, since I had come from AP calc) and so we covered things like the Bolzano-Wierstrass Theorem, basic theorems in calculus and their proofs/applications, and the Lebesgue measure. Here, we really just talked about the measure in its connection with Riemann integrability, not actually discussing measure theory itself.

It wasn't until the spring semester where he showed us the topological/metric point of view and the theme was essentially proving all our theorems from the fall semester in the more general multidimensional setting.

If you feel like your analysis class wasn't deep enough, you can always go out and pick up the well-known books and read through them yourself! I personally learned measure theory from Royden's text and found that it smoothly fit in with what I already knew.
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