Need Guidance for engineer taking rigorous analysis course for first time

Question

I will be taking analysis course in a month from now. Topics are given below. I am doing engineering and had been through calculus courses but nothing like sort of analysis before. Many of my friends who are engineers themselves have warned me about the difficulty and abstraction level of this course. Can some one suggest what should i do in order to learn it properly. I will be sitting with maths majors and donot want to look dumb. Any books for self study or whatever will be appreciated.Any tips or help thanks..

Thanks

MAL411 Topics in REAL ANALYSIS, 4 (3-1-0)

Course contents : Metric spaces, completeness, connectedness, compactness, Heine-Borel theorem, totally bounded sets, finite intersection property, completeness of R^n, Banach fixed point theorem, perfect sets, the Cantor set. Continuous functions, relation with connectedness and compactness, discontinuity, uniform continuous functions, monotone functions, Absolutely continuous functions, total variation and functions of bounded variations. Differentiability and its properties, mean value theorem, Taylor's theorem, Riemann integral with properties and characterization, improper integral, Gamma function, Directional derivative, Partial derivative, Derivative as a linear transformation, Inverse and Implicit function theorems, multiple integration, Change of variables. Sequence and series of real numbers, point wise convergence, Fejer's theorem, power series and Fourier series, uniform convergence and its relation with continuity, differentiability and inerrability, Weierstrass approximation theorem, Equi-continuous family, Arzela-Ascoli theorem.

Answer 1

I recommend three books for self reading

1. Real Analysis and Applications by Donsig and Davison

This book is extremely well geared for engineers. It also has real applications at the end to motivate the theory such as dynamical systems, wavelet transform, convex optimization. But it does not go into metric spaces, topological properties etc, right away. If you are going to read this book, read it fast so you get a feeling of the basic concepts, then move onto another book that's more geared towards math students.

2. Understanding Analysis by Stephen Abbott

One of if not the clearest text ever written for an introduction to real analysis and covers most of the topics there. All the problems has solution at the back of the book. Just a very very clear book, great for self study.

3. Real Mathematical Analysis by Charles C. Pugh

Without a doubt covers all the topic on there, going quite deeply into topological properties and going well beyond all the topic that are there. Equicontinuity and Arzela-Ascoli theorem are heavily covered. That being said, this book is terse and uses unconventional terminologies, which could cause irreparable damage if studied alone. Nonetheless, this book is going to be the best fit for the course you are about to take.

Some other modern text such as Real Analysis by Terence Tao is also great. Joseph Muscat (classmate of Terence Tao) also has a book called Functional Analysis which I think is also great and rich with examples. In fact, functional analysis is what engineers actually use in practice. Bartle and Sherbert's Real Analysis is also great but a bit long winded and covers metric spaces last.

Matt Pon's Real analysis for the undergrad is also great, wonderful book rivals clarity of Understanding Analysis but his way of covering measure theory is unconventional.

Answer 2

Step 1: get comfortable with how to write proofs. This is not so easy, and really must come with lots of practice and experience. Reading guides (like here and here) with examples and trying to write very simple proofs yourself (and comparing yours with theirs) will be very helpful. My experience is that in a first course in analysis, many students have no proof writing experience, and this is not addressed by professors despite the fact that the entire course is proof based.

Step 2: intuition and concepts are very important to have. Often these courses are taught in a very definition $\to$ theorem $\to$ definition $\to$ theorem $\cdots$ manner, and the proofs are presented as in Rudin's book: lots of (correct!) but otherwise unmotivated (to a non-analyst, like a student) steps which get you from $A$ to $B$... somehow. This isn't how people come up with proofs, though. They use intuition and concepts to come up with the idea and then carefully translate these using the formalization into a rigorous proof. So, my advice here is to use many different resources (not just Rudin) to really understand what's going on. It's not so hard to "understand" a proof by reading through it line by line and verifying what's said is true, but it's another thing to generate proofs on your own, and this requires a degree of understanding gotten only from finding a source of information that "jives" with you and your way of learning/thinking.