Talk:Real analysis

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)

I am wondering, what is the purpose of having this page with not even an introduction of what it really is? lucarioic 03:50 19 Oct 2012 (UTC)

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)

Correct redirect?

Real function redirects here. Is this right? I was actually looking for the article Real part when I got here. My english skills are too limited in this area, but it seemed like a reasonable search string for that article. Is there any point in adding some kind of see also... or for other uses... section? -- Asked by... 83.227.131.25 (talk) 13:31, 31 March 2009 (UTC)

I've never heard that use of "real function", but I can see the logic behind it.-Wafulz (talk) 15:09, 31 March 2009 (UTC)

Thanks for your answer. If it doesn't make sence to native english speakers, forget it – it was probably just a matter of bad translation or misunderstanding from my side. I've also seen it written as $\operatorname {Re} (z)$, which kind of looks like a function. Cheers, says 83.227.131.25 (talk) 15:54, 31 March 2009 (UTC)

Set or field of real numbers?

An editor changed the description of real analysis from dealing with the "set" of real numbers to the "field" of real numbers. I question whether this is an improvement. Of course, I do not deny that the real numbers form an algebraic field, but the topic of real analysis is not specifically algebraic. One might as well say that it deals with the "space" of real numbers. What do you folks think? TomS TDotO (talk) 09:21, 15 August 2010 (UTC)

Does the word "set" even need to be in their? "Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the real numbers" — Preceding unsigned comment added by Brad7777 (talk • contribs) 15:49, 23 October 2011 (UTC)

@(TomS TDotO): Real analysis is not algebraic? Let's see, with the set of real numbers we can talk about unions and intersections, and that's about all (and incidentally, we can't tell them apart from the set of complex numbers). If we want to do some fancy stuff like add or multiply them, then we've got to consider the field of real numbers. On the other hand, I still agree with you that just because it's true, doesn't mean we have to say it in the lead, and I'm going to implement Brad's suggestion to just call them "the real numbers". — ChalkboardCowboy^[T] 20:20, 6 July 2012 (UTC)

I like the idea of field because it emphasizes more structure and also provides a relationship to other branches of mathematics. But perhaps it would be more accessible or simple to just say that it deals with the "real numbers". Cazort (talk) 17:39, 18 October 2012 (UTC)

Proposed merge

In analogy with function of a complex variable and complex-valued function redirecting to complex analysis, it seems natural that the two stub articles function of a real variable and real-valued function should be merged to this article. Comments? Isheden (talk) 17:47, 30 July 2012 (UTC)

This seems like a problem with the complex analysis article: if I didn't know what a complex-valued function was I don't think I would be able to figure it out from the complex analysis article. Same thing with function of a complex variable. There is a little that can be said in these other articles that probably shouldn't be in this article, so I think they should stay separate. Jbeyerl (talk) 12:50, 1 September 2012 (UTC)

I don't like the idea of merging and redirecting any of these things. Complex analysis and real analysis are fields, and complex-valued functions are objects of study in these fields. Both seem notable as subjects in and of their own. Cazort (talk) 17:33, 18 October 2012 (UTC)

• Agree with the merge --- function of a real variable and real-valued function are definitions only and trivial definitions. We can't have them as stubs. Also, when they are objects of study, we are talking about Real analysis or Mathematical analysis.

Thanks, Anand ^{(talk page)} 02:38, 7 May 2013 (UTC)

expert-subject tag?

This article was tagged as needing attention from an expert, with no specific reason expressed. I read the article and didn't see anything glaringly inaccurate. What was the issue that needed to be addressed? Jbeyerl (talk) 12:56, 1 September 2012 (UTC)

"The foundation of real analysis"?

At the Key concepts section, the article claims that "[t]he foundation of real analysis is the construction of the real numbers from the rational numbers." I understand this pedagogically, but might this be potentially misleading? My understanding is that what enables one to do useful analysis is that the reals form an ordered field that's complete (both in the sense of Cauchy-completeness and Dedekind-completeness). Indeed, from the page Real_number, up to field-isomorphism, the reals are unique as the only totally ordered field, complete and Archimedean with respect to that ordering.

From this view, it's this field- and order-structure of $\mathbb {R}$ that is central. The importance of the constructions of $\mathbb {R}$ from $\mathbb {Q}$ (via Dedekind cuts or equivalence classes of Cauchy sequences of rationals or nested intervals with rational endpoints or whatever) is that it concretely demonstrates the existence of an object with these desired properties. (Subject, of course, to acceptance of the logical consistency of the rationals in the first place.) There are, however, other ways of constructing $\mathbb {R}$.

For example, in Numbers by Ebbinghaus et al., Springer-Verlag, 1991, ISBN 0-387-97497-0 (itself a translation of the second edition of the German Zahlen, 1988, Springer Verlag), Section 2.5 describes an axiomatic definition of the reals. (See also Construction_of_the_real_numbers#Synthetic_approach and Tarski's_axiomatization_of_the_reals.) This vantage point views the reals ab ovo, and we then show that we can recover $\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R}$ with their usual respective properties. In Chapter 13 (Ibid.), the author outlines John Conway's "game" method for constructing $\mathbb {R}$ "in a single step."

I would provisionally propose alternate wording like the following, borrowing from Real_number:

The foundation of real analysis is the existence of a Archimedean complete totally ordered field $(\mathbb {R} ,+,\cdot ,<)$, unique up to isomorphism of ordered fields. The standard method of verifying the existence of such a structure is constructing the real numbers from the rational numbers, $\mathbb {Q}$. Uniqueness typically follows from a separate argument.

I am open to suggestions about whether the original wording merits changing in the first place, whether the above proposed wording would indeed constitute an improvement, and what other changes might further improve this section.

As an independent matter, Numbers cited above may prove useful to authors who wish to expand this article, especially for content regarding the historical development of real analysis. — Preceding unsigned comment added by 71.65.211.107 (talk) 07:12, 14 October 2012 (UTC)

I think it is subjective to claim that any one approach is "the foundation" of a subject. I think it's more fitting with WP:NPOV to write things like: "The subject of real analysis can be developed from the construction of the real numbers from the rational numbers." But this doesn't reflect the historical development of the subject--many results came before the construction was carried out. And one can work in the subject without ever constructing the real numbers, accepting them as a given. I also want to point out that mathematical constructivists, rejecting the axiom of choice would have a different perspective--yet there is still that can be done within the field of real analysis in a constructivist framework. It's important that we don't present a single POV as the only one, or give undue weight to one. Cazort (talk) 17:36, 18 October 2012 (UTC)

I agree that "foundation" is probably a problematic word here, whether as in the original or in the rewording. Does this mean logical foundation, historical foundation, some hybrid between the two, or something else entirely?

That said, if there is to be a "Key Concepts" section, I think it's defensible to on neutrality grounds to emphasize those models of $\mathbb {R}$ which lead to the subsequently-enumerated useful, familiar theorems most likely to be accessible to a lay audience: Bolzano-Weierstrass, Heine-Borel, intermediate value theorem, etc.

I'm not sufficiently knowledgeable about constructivist models to know which theorems, if any, might break down in constructivist or other cases. But if detailing all the various models or methods for constructing $\mathbb {R}$ would be a digression for the sake of NPOV, then might it simply be more sensible to append constructing the reals to the See also section and remove this sentence entirely? Perhaps a wholesale reorganization of this section would be merited. 71.65.211.107 (talk) 17:57, 19 October 2012 (UTC)

General Metric Spaces

In some parts of this page, for example the definition of Uniform Continuity, definitions are constructed in the setting of general metric spaces (X,d). Given that this article is purely about the real analysis, would it not make more sense to define everything less abstractly, in terms of the real numbers rather than a general set X? Someone looking for information on abstract analysis would surely go the the page Mathematical Analysis or, in this specific case, Uniform Continuity. EdwardRussell (talk) 17:13, 11 November 2013 (UTC)

I deleted the metric space material; I added it in accidentally when I wrote that section. Thanks! Brirush (talk) 20:27, 11 November 2013 (UTC)