A very open ended question about Real Analysis

In summary, most people say that linear algebra and intro proofs courses are relatively easy, while abstract algebra and Analysis courses are among the more difficult ones in a mathematics major. People also say it helps to think of mathematical objects in terms of their properties instead of metaphors and approximations in the real world. Some people find real analysis to be hard, but most people find it satisfying once they "get it".
  • #1
anonymity
163
0
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).
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #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).
 
  • #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.
 
  • #6
it helps if you learn from a good book, widespread use of rudin in my opinion makes the subject harder than necessary.
 
  • #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.
 
  • #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.
 
  • #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.
 
  • #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.
 
  • #11
snipez90 said:
Those people may realize right then that college math is not as awesome as they thought it would be.QUOTE]

lol. Love it.

I guess I should just be happy that I get to take an introductory proofs class beforehand (and linear algebra too) to practice my proof writing..then I can dedicate the majority of my time to the concepts and depth.

Just want to thank everyone again for chimming in with their opinions.
 
  • #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).
 

FAQ: A very open ended question about Real Analysis

What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers and the properties of functions and sequences of real numbers.

What are the main topics covered in Real Analysis?

The main topics covered in Real Analysis include limits, continuity, differentiation, integration, and series. It also includes the study of metric spaces, topological spaces, and measure theory.

Why is Real Analysis important in scientific research?

Real Analysis is important in scientific research because it provides a rigorous foundation for understanding and analyzing mathematical models and data. It also helps in developing and proving theorems and formulas used in various scientific fields.

How is Real Analysis related to other branches of mathematics?

Real Analysis is closely related to other branches of mathematics such as calculus, algebra, and geometry. It also has applications in areas such as physics, engineering, economics, and computer science.

What are some real-life applications of Real Analysis?

Real Analysis has many real-life applications, including in finance, where it is used to model and analyze the stock market and interest rates. It is also used in physics to study motion and in engineering to design and optimize structures and systems. Additionally, it is used in data analysis and machine learning to develop algorithms and models.

Similar threads

Replies
11
Views
3K
Replies
10
Views
2K
Replies
8
Views
3K
Replies
9
Views
580
Replies
4
Views
7K
Replies
11
Views
2K
Replies
7
Views
4K
Back
Top