Talk:Real analysis

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[edit] Initial comments

Isn't this suficiently dealt with on the mathematical analysis page? Why is a separate page needed? -Stuart

This page's structure - explaining what sequence a book would introduce the subtopics - doesn't really make an encyclopedia entry i don't think... Enochlau 17:29, 21 Apr 2005 (UTC)

Old page - we do things differently now. Charles Matthews 17:38, 21 Apr 2005 (UTC)
Yes this page needs to be rewritten ;-) Paul August ☎ 17:54, Apr 21, 2005 (UTC)

I would like to see some examples of where this sort of extremely abstract mathematics is used in the field, particularly with links to the appropriate astrophysics and quantum theory pages. -Eliezer Kanal 11:24 PM, Oct 15, 2005

Well, I do think examples would be nice, but real analysis is not just used in those fields, and it can be very limiting to say "this is what it's for". Analysis is a stepping stone to a lot of other mathematics. Higher level study of probability that's used in mathematical finance requires analysis, for example. I think some of the "real-life usages" in other articles have made those articles worse. Tristanreid 19:20, 16 October 2005 (UTC)

I just wasn't sure if it was proper to state that the Real's are constructed by Cauchy sequences, when the basis for their definition relies on the completeness axiom and concept of supremum and infimum. If there is another approach to defining a Cauchy sequence I appologize.

I would say that the least upper bound property is far and away the most important property of the real numbers, not the properties of the absolute value function as stated in this article.

I added a little bit based on the reference which I also added - I just took my Intro to Real Analysis final on Friday :) If I've done an improper edit, please forgive me because I have only just begun trying to contribute. Anastas5425 21:04, 23 April 2007 (UTC)

of course real numbers can be defined as an ordered field of characteristic 0, that has least upperbound property. so least upperbound property is of course the most important. Jeroje 20:37, 14 July 2007 (UTC)jeroje

This article is appallingly bad...I don't think it's already covered in mathematical analysis since that article (as it should) points the reader to a whole big list of specific sub-fields of mathematical analysis. This article should look more like non-standard analysis, complex analysis, or at least functional analysis. A good start would be, say, anything that doesn't look like a course outline. Specific techniques like epsilon-delta proofs should be mentioned, a history section should exist, and lots of other stuff - I'll be starting on it tomorrow. Meowist 11:44, 21 June 2007 (UTC)

As I mention below, I've redone major parts of the article, and I've tried to accomodate as many of the comments on here as possible, but there are a couple of things Meowist suggested that I haven't done, and I thought I should explain why.
Writing an article on real analysis is a bit like writing an article on non-hybrid cars. For a long time, all analysis was real analysis, but recently other techniques have emerged, and it became meaningful to talk about real analysis as being different to complex or functional analysis. You can't really talk about the history of real analysis as being distinct from the history of analysis as a whole, any more than you can separate the history of the car from the non-hybrid car.
Also, including a section about epsilon-delta proofs seemed a little inappropriate, as this is a technique that is not specific to real analysis, but is common to all areas of analysis. It would be like putting a section on headlamps in the hypothetical article on non-hybrid cars.
Anyway, I hope that explains some of my editorial choices, and I apologise for the lousy metaphor. James pic 13:19, 26 June 2007 (UTC)

[edit] Overhaul

I've taken the liberty of redoing large parts of this article. Much of the material was about analysis in general, and not specific to real analysis, and a large portion of it simply outlined the syllabus of a first course in real analysis. I've tried as best I can to keep the stuff that's relevant, to remove the stuff that's better dealt with elsewhere (or just plain unsuitable), and to put the material into context, although obviously more work will be needed. This article was an embarassment, and I can't claim to have fixed it, but I've tried to clean it up, and make it more maintainable. Please edit ruthlessly. James pic 12:09, 26 June 2007 (UTC)

I am very pleased with your rewrite. In response to your comments, in hindsight, a history section would perhaps mirror too closely a proper history section in Mathematical Analysis (the one there is hideous). If analysis is likened to a tree, then real analysis is the stem from which everything else forks. What I had in mind for the section with epsilon-delta proofs was simply an example of analysis concepts being used in a proof of real-analyis-type statement - nothing like an actually famous theorem, just some small slightly non-obvious claim. Here's an example of what I'd put in: Proving the limit of \sqrt(n^2+n)-n as n goes to infinity is 1/2. I'd use the continuity of sqrt(), and some epsilon-delta. What do you think about the general idea of including an example or/and this one in particular? Meowist 02:29, 27 June 2007 (UTC)

Certainly, an example like the one you suggested could be useful. As for choice of example, I'd probably go with something that demonstrates the character of real analysis, such as the monotone convergence theorem - the proof is a simple application of the least upper bound property and some simple delta-epsilonics. Obviously though, it's not my article; make whatever changes you feel are appropriate. James pic 12:54, 27 June 2007 (UTC)