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A very open ended question about Real Analysis

  1. Aug 21, 2011 #1
    I was hoping to get some personal opinions regarding the first round (two semester sequence) of undergraduate analysis.

    How difficult do YOU think that these classes are? Use comparisons as you feel fit (linear algebra, intro proofs course, abstract algebra, etc).

    (I do realize how general, even obscure, this question is. I looking for opinions, not definite answers).
  2. jcsd
  3. Aug 21, 2011 #2


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    It was long ago (1958) that I took my first such course. It was called Advanced Calculus then and taught out of a book by Louis Brand. It was the first rigorous course I had encountered and I found it to be very difficult. I made it through the course and a year later I began to "get" it. At that point I also began to understand the notion of mathematical maturity. Sometimes it just takes a while. Your mileage may vary.
  4. Aug 22, 2011 #3


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    Typically, such things as Linear Algebra and intro proofs are relatively easy (for a person who has done very well in previous math courses) while Abstract Algebra and Analysis are are among the more difficult courses in a mathematics major.
  5. Aug 22, 2011 #4
    This seemed to be the general consensus, I just wanted to ask directly.

    When you say linear algebra is generally on the easy side, are you talking about the typical junior level proof based class, or an applied class?

    I am taking set theory (intro proofs) and linear algebra (proof class) this year, and intro real analysis next year (my junior year). I really want to put my best foot forward in real analysis next year and (futile as it may be) I would like to do what I can now to prepare for it (in a sense -- I'm not getting a book now or anything, I just want to be ready to really understand the subject when next year comes around). That said, is there really anything I can do, aside from learning to fluently write and understand proofs? (I do realize the drastic turn that I made as far as the direction of where this thread was going - I am sorry).
  6. Aug 22, 2011 #5
    A notion that may be helpful in those more abstract courses (and this is admittedly not any easier to do just because you've heard it said outright) is to think of everything in terms of their properties, instead of in terms of metaphors and approximations in the real world.

    Developing a good metaphor is helpful for some purposes, but the problem is that they only go so far, and too good a mnemonic can be a stumbling block when you reach that point where the mathematical object that you're studying departs from your visualization of it. This is a problem that has held mathematicians back for centuries, making them see these departures as problematic or "paradoxes," when in fact the only shortcoming was that they were hung up on an insufficient metaphor.

    And again, metaphors are very useful because they can provide intuitive explanations for difficult concepts. Just remember that if your metaphor fails, it's not something to be worried about, it just means you're developing a deeper understanding of the object.
  7. Aug 23, 2011 #6


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    it helps if you learn from a good book, widespread use of rudin in my opinion makes the subject harder than necessary.
  8. Aug 24, 2011 #7
    My school uses Apostol (atleast they are currently), and I don't expect that will change for next year. Amazon seems to receive him well; I really couldn't say though.
  9. Aug 24, 2011 #8
    I found analysis highly satisfying when I "got it". It felt really good to write out a proof and feel confident that it was air-tight and complete. Unfortunately, the effort required to "get it" was huge for me and I found the class very difficult and time consuming.

    Fortunately I took it during the summer with only two other students in my class and we were able to work together and help each other with our understanding and that made all the difference in the world. If I had taken it during a normal semester I would have had a very difficult time balancing it with my other courses. In the summer I only had to balance it with research.
  10. Aug 24, 2011 #9
    I don't know why everyone finds Real analysis to be hard.

    Literally it is 1 definition and playing around with stuff that is obvious most of the time. Certainly, the triangle inequality and having other bunch of inequality in your mind isn't hard.
  11. Aug 24, 2011 #10
    It's true that basic real analysis is not that hard once you have a firm grasp of the epsilon delta definition, but it's the first class most aspiring math majors take. Those people may realize right then that college math is not as awesome as they thought it would be.

    Obviously real analysis gets really deep pretty quickly. For instance, analysis of PDEs is pretty hard, and even a rigorous treatment of the basics require at least measure theoretic ideas, but often much more. Carleson's Theorem is also pretty hard.
  12. Aug 24, 2011 #11
  13. Aug 24, 2011 #12
    I took algebra first and on the one hand I'd like to say I found it more intuitive, on the other hand I liked it significantly more and studied out of the book on my own a good bit, whereas real analysis felt more like a chore and I was bad about doing all of the course work and actually studying.

    I got an A in algebra, A- in real; as to which is harder, I'd say that they're probably similar in terms of difficulty, but most people seem to have a preference for one over the other. Certainly both areas get deep very quickly, in my second semester of abstract we were covering Galois theory and after that you can branch out into commutative algebra (which leads into algebraic geometry) or non-commutative stuff or algebraic topology stuff, algebraic number theory etc.

    I've been told that people tend to break up into camps of having a preference for more discrete mathematics such as algebra/number theory/logic and diff geometry/analysis/topology (though of course many areas, such as algebraic geometry, have quite a bit of overlap). As to whether that's totally true; it seems to be, but there are definitely plenty of exceptions. There has been tons of fruitful work in crossover areas (non-standard analysis stuff, for instance, which I believe has been applied to some problems in the study of PDEs).
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