Background needed for analysis.

[iamthegelo]

Hi,

I have completed my University's Calc 1-3 courses and I also have done some Linear Algebra and ODE's. I thought these were enough to start reading an Analysis book like Shilov's but I found the first part confusing like sets, metric spaces, etc. I was wondering what are the pre-requisites to learn analysis? Or maybe, are there other books that are simpler? Maybe something more introductory? I was leaning towards getting Yet Another Introduction to Analysis by Bryant or Introduction to Real Analysis by Schramm.

Thanks.

 

[PieceOfPi]

How was your Calc 1-3 like? Did the class stress more on proofs and theories of calculus rather than computations and applications? For example, did your calculus talked a lot about epsilon-delta stuff when you talked about limits and continuity? If you want to study analysis, it is much better to be exposed to rigorous calculus before you actually study it. If you haven't seen rigorous calculus, I suggest you to read Kenneth Ross's Elementary Analysis: The Theory of Calculus. This text covers all the theory that are omitted from standard single-variable calculus class, so you will be more prepared for the analysis book that you are trying to read. I think this is a great textbook--it's well-written, the price is reasonable, and it has solutions/hints to some of the problems, so it's a nice book for self-studying. This text also has optional sections on topology and metric space, so I would read them as well. I took analysis course without reading those optional sections, and now I wish I had since I didn't do so well in the course :(

 

[iamthegelo]

My courses were for applications (engineering major) rather than theory. I was also looking at the book by Ross and will get it and restart from there. I've had problems with integration and would like to really avoid that weakness as soon as possible. I think rigorously studying it would be best in order to get rid of the weakness.

Thanks.

 

[snipez90]

The problem is that you have probably never seen the motivation for abstract concepts such as a metric space. A metric space is a set that generalizes the notion of "distance" and serves as a measure of the degree of closeness between two points in the set. In a rigorous calculus course, you work with the one-dimensional case of the euclidean distance, i.e., the real numbers with distance function d(x,y) = |x-y|. This metric is of course, extremely intuitive, since you can "visualize" the real line.

The properties that the distance function of a metric space have to satisfy are also pretty intuitive if you think about how the concept of distance should be generalized. We clearly don't want negative valued distances. The "distance" between two points of the set should be 0 if they are the same point (the converse should also make sense). Clearly, the order of the arguments of the distance function don't matter, since the "distance" between two points should not change if we consider one point before the other. Finally, the inequality that must be satisfied gives the notion of "transitivity" in the metric space. If x, y, z are elements of our underlying set, then if x and y are "close" and y and z are "close", then x should be "close" to z.

A rigorous calculus book will provide motivation, but you will almost certainly see a repeat of many of the epsilon-delta proofs of limits or continuity in the metric space chapter. Instead of working through an entire calculus text, try to see if you can come back to metric spaces after you gain a good intuitive sense of the epsilon-delta definition of limits.

Also, note that learning the basic theories of integration is probably not going to help you much in computing integrals. The proofs of the integration by parts method and the substitution method are pretty simple (the latter relying on the fundamental theorem of calculus). The fundamental theorem of calculus itself can be proved once you learn the theory of Riemann integration or Darboux integration. However, it is unlikely that learning either of these theories will improve how well you integrate. Psychologically, it may help, but who knows.

 

[blerg]

For someone in your situation I would recommend Introduction to Real Analysis by Bartle. Its a great transition from computational calculus to a rigorous analysis course (a la Rudin)

 

[Landau]

Spivak's Calculus provides, despites it name, a very good intro to analysis (in one variable) with lengthy and crystal clear explanations. After that, you're more than good to go for Rudin, Shilov, etc.

 

[n!kofeyn]

I agree with trying out Calculus by Michael Spivak. Also, Calculus by Tom Apostol is a very rigorous calculus text, and his Mathematical Analysis book is a great analysis book. Although, his books are a little dry, but are very clean and contain tons of information. You might be uncomfortable with proving things, so I would recommend Analysis: With an Introduction to Proof by Steven Lay. It is a fantastic introduction to rigorous calculus and has some basic analysis.

I like to discourage Rudin. I don't even find the book to be a good reference, and always turn to Apostol's Mathematical Analysis or another reference. Also, Advanced Calculus by Creighton Buck is old gem. His style is rather unique. I'll have to check out Ross' book, as it looks good as well.

 

[iamthegelo]

Do I need Topology?

 

[n!kofeyn]

Do I need Topology?

Well most introductory analysis books teach some basic metric space topology, and that is all you really need in the beginning. So there is no need to pick up an extra topology book. All this just depends on what you want to accomplish. If you want a thorough review of rigorous calculus, then I would go with the Spivak, Apostol, or Courant calculus books. If you want to start learning analysis, which sounds like what you asked, then I would go with Analysis by Steven Lay. It contains an introduction to logic, functions, and techniques of proofs, which will be of great use to you. He then follows this in the second half of the book with a great introduction to analysis, and the book will flow nicely into a more intermediate analysis book when you're done.

 

[jasonRF]

Well most introductory analysis books teach some basic metric space topology, and that is all you really need in the beginning. So there is no need to pick up an extra topology book. All this just depends on what you want to accomplish. If you want a thorough review of rigorous calculus, then I would go with the Spivak, Apostol, or Courant calculus books. If you want to start learning analysis, which sounds like what you asked, then I would go with Analysis by Steven Lay. It contains an introduction to logic, functions, and techniques of proofs, which will be of great use to you. He then follows this in the second half of the book with a great introduction to analysis, and the book will flow nicely into a more intermediate analysis book when you're done.

I second the recommendation of Lay's analysis: with an introduction to proof. It does a nice job of teaching about proofs, and basic analysis. I picked up a used copy of the second edition cheap, and have worked through 5.5 chapters so far - it is worth the effort and within reach of us novices who are studying on our own.