# Apostol 1 vs Ross's Elementary Analysis

1. Oct 14, 2012

Hi everyone,

I left college many years ago and now I've forgotten a lot about multivariate calculus. I've also never taken a real analysis course of any kind before. Now I'm planning to go back to school for a graduate degree in econ. The school says that I must have good skills in real analysis to survive those econ core courses because exercises are all proof-based.

I have with me now Apostol 1 and Kenneth A. Ross's "Elementary Analysis: The Theory of Calculus". Reading 3 chapters from both texts, I'm surprised that Ross is an easier read than Apostol 1, even though Apostol is about calculus and Ross about intro real analysis.

Questions:
1) Is calculus (single and multivariable) a pre-requisite for real analysis? On Wikebook course in real analysis, it says that to learn real analysis requires no background in calculus!!! I'm not sure if this statement is correct or not.

2) Should I go straight to Ross or stick with Apostol 1, given that I don't have much time left to prepare before I go to grad school in econ?

2) On the UC Berkeley econ dept website, there is a course called Econ 204, which is basically a math camp for PhD econ students, here, http://elsa.berkeley.edu/users/cshannon/e204_11.html

I've seen a few lectures from this Econ 204 course and found that they are pretty hard material. However, the syllabus says that students can get a waive provided they've taken a course in real analysis comparable to UC Berkeley's Math 104. I then googled for UC Berkeley's Math 104, http://math.berkeley.edu/~chr/104.S12/index.html , and found that they used Kenneth Ross's text for the course.

My question is, why is it that the instructor consider a course closely based on Ross's text a substitute for her Econ 204 course when the latter touches on far advanced topics like optimization, differential equations, etc...? Does she imply that her course, i..e Econ 204 stresses proof writing and not on topics?

Thank you very much.

Best,

2. Oct 14, 2012

### chiro

Hey adum and welcome to the forums.

The answer to 1 is an emphatic yes. Real analysis formalizes calculus and gets really specific with regard to what the concepts mean and how to prove them.

Again the point of analysis is to make the machinery of calculus work and also to go to the core of mathematics but introducing these concepts so that not only can you check that all the calculus and other limit based stuff actually works, and that things "make sense" like converging, existing, and being unique, but you can also take arbitrary examples and prove these properties yourself using the existing machinery.

If you haven't got a solid grasp of linear algebra, norms, inner products, and the application of these to limits, convergence, and proving a variety of results then if the course is what they say it is, you will be in a precarious situation.

The calculus stuff you have taken is meant to give the intuition behind what's going on geometrically, visually, and otherwise but the analysis will assume that you have the intuition to follow the symbolic transformations and arguments and doing this is just a specific kind of skill just like using a lathe vs using a jigsaw are two specific skills in carpentry.

3. Oct 14, 2012

Thank you Chiro.

Do you think Kenneth A. Ross's intro analysis text, or Apostol 1 is a sufficient background for such texts as baby Rudin or Apostol's Mathemtical Analysis.

4. Oct 15, 2012

### chiro

I can't really advise you on the choice of textbook for either of these cases or in relation to what is covered in the course.

I'm guessing (and this is speculation) that if you have worked through one analysis text, you should be OK.

Again some of the main bread and butter principles of analysis are convergence and limit proofs along with the use of a wide variety of results involving norms of all kinds and also matrices and their properties. You also need to consider the relation between norms and inner products if an inner product exists.

Any analysis book should cover these in detail and you will have to prove a variety of results.

I'm not a pure mathematician so I'm not going to go beyond this response.

5. Oct 15, 2012

### Studiot

The Professor of Mathematics at the London School of Economics has written a book just for you.

Calculus by K G Binmore.