1. ## What is analysis?

If someone had never taken Analysis before, but asked you what it was, what would you say?

A friend of mine said that you take everything you ever learned about math, and then relearn it, but this time, leaving NO stone left unturned. Is this a good way to look at it?

2. When I was coming up, "analysis" just meant Calculus. Thus "complex analysis" is the Calculus of complex-valued functions, etc. "Real analysis" deals in an involved way with the topology of the real numbers in order to rigorously construct the apparatus of Calculus, etc... If one says that Liouville used "analytical methods" to prove the existence of transcendental numbers whereas Cantor used set theoretic arguments, one is referring to the fact that Liouville's proof uses the Mean Value Theorem for Derivitives from Calculus, etc.

3. Originally Posted by AlephZero
When I was coming up, "analysis" just meant Calculus. Thus "complex analysis" is the Calculus of complex-valued functions, etc. "Real analysis" deals in an involved way with the topology of the real numbers in order to rigorously construct the apparatus of Calculus, etc... If one says that Liouville used "analytical methods" to prove the existence of transcendental numbers whereas Cantor used set theoretic arguments, one is referring to the fact that Liouville's proof uses the Mean Value Theorem for Derivitives from Calculus, etc.
This pretty much the explanation I give as well. "An advanced, proof based treatment of all you learned in calculus and then some." Though this changes depending on who I am talking to of course.

4. Originally Posted by putnam120
...Though this changes depending on who I am talking to of course.
You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

Two questions:

1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

2. What is the absolute best book for beggining analysis (real)?

5. Originally Posted by VonNemo19
You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

Two questions:

1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

2. What is the absolute best book for beggining analysis (real)?
1. I can't think offhand of a reason why you'd need either ODE or linear algebra for real analysis, although you might want to take linear algebra first because it's usually an easier course. ODE requires some linear algebra (Wronskians, etc.), so take linear before that.

2. There's probably never going to be a concensus on what's the "best" book on real analysis. I'm partial to Richard R. Goldberg's Methods of Real Analysis (2nd ed., John Wiley & Sons, 1976), because that's the one I learned it from, though I'm a broken down old dude.

6. Originally Posted by VonNemo19
...Would it be better to take DEs or Linear algebra before analysis? Or the other way around?...
What about complex vs. real analysis in terms of which should be taken first?

7. Originally Posted by McScruffy
What about complex vs. real analysis in terms of which should be taken first?
I don't think it makes an awful lot of difference. I guess I would say take real analysis first, because it deals intensively with strict definitions of continuity, sequences, and series. That way, when issues of continuity and such come up in complex analysis, it will be really clear to you what's going on. In my experience, the really hard stuff in complex analysis is in two areas: contour integration and residue theory. I suppose you'd be somewhat better prepared for each after having the kind of ultra-rigorous integration and infinite series work one usually gets out of a real analysis course.

8. Originally Posted by VonNemo19
If someone had never taken Analysis before, but asked you what it was, what would you say?

A friend of mine said that you take everything you ever learned about math, and then relearn it, but this time, leaving NO stone left unturned. Is this a good way to look at it?
Analysis is the rigorous study limiting processes over various topological spaces, for real analysis these spaces will be the reals and assorted function spaces over the reals, for complex analysis ...

All you learned in calculus will be re-done in real analysis but without missing out the basic assumptions that you probably never noticed that you had to make.

The main reason that analysis emerged in its present form was to make sense of Fourier series (and to cure the problems that (George) Bishop Berkeley identified in "The Analyst"). You might be interested in "A Radical Approach to Real Analysis" by David Bressoud which discusses the emergence of analysis.

Note before the emergence of modern analysis in the early 19th century the term analysis referred to calculus.

CB

9. Originally Posted by VonNemo19
You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

Two questions:

1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

2. What is the absolute best book for beggining analysis (real)?
1. I personally took Linear algebra before analysis but that was mainly because it was a prereq. for the class. Really I found it unnecessary. However, it would have been nice to know Linear algebra if we had studied functions $f:\mathbb{R}^n\to\mathbb{R}^m$. As for DEs I found that some results in my DE classes made more sense after taking analysis, but that's just me. Though I would suggest taking an "introductory" DE class after your initial calculus sequence.

2. This really depends on the person. I personally liked Rudin (my first analysis book) once I got past the topology chapter. However, I know people who hate it.

10. Originally Posted by VonNemo19
You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

Two questions:

1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

2. What is the absolute best book for beggining analysis (real)?
On your first question, I would recommend taking both before real analysis. Real analysis is a sophisticated subject and the more mathematically mature you are the better you will be prepared. Just my two cents.

11. Originally Posted by Danny
On your first question, I would recommend taking both before real analysis. Real analysis is a sophisticated subject and the more mathematically mature you are the better you will be prepared. Just my two cents.
I generally agree with that statement.
In my experience, the main value in undergraduate ‘Complex Variables’ lies in its teaching the nature of a complex numbers. (That is not done completely in any other undergraduate course.)
Now turn to Linear Algebra. In Linear Algebra one learns about basis. One learns to use matrices to for a many different purposes.

Both of those out comes will enhance your study of real analysis.

12. Thanks guys. (and girls if there are any)

13. Here is a good description of the "point" of doing real analysis: Here.

14. Originally Posted by Sampras
Here is a good description of the "point" of doing real analysis: Here.

Very good!