Talk:Real Analysis

I feel that Real analysis now contains quite a bit of stuff on quite a few topics. However, the book needs some clean-up and some major re-organisation to make it more reader-friendly (though new material is also welcome, particularly on multivariable analysis and Fourier analysis).

SPat (talk) 08:46, 23 May 2008 (UTC)

Also, these are suggestions from previous authors (moved here from the contents page)

 Other suggested topics for inclusion 

Since the goal here is to put calculus on a solid footing, I am going to add background, so that we develop the concept of a number first, and work to functions more slowly and methodically, to include Heine-Borel, Weierstrass, etc.

Things that seem to fit in this context:
 * (Basic) functional analysis:
 * Uniform Convergence, function spaces
 * Arzela-Ascoli Theorem
 * Stone-Weierstrass Theorem
 * Riemann-Stieltjes integrals and bounded variation
 * Measure theory:
 * Measure theory/Lebesgue integrals
 * Generalized function (distributions) theory
 * Some basic harmonic analysis (Fourier series and transforms).

However, both functional analysis and measure theory could do with their own Wikibooks

Things that might better be in Set Theory:
 * Infinite sets and cardinality

Things that might better be in Topology:
 * Introduction to different concepts of space: topological, metric, normed, inner product
 * Basic topology: accumulation points, closure, interior, boundary, convergence of sequences, in each case with discussion of the type of space the concept is appropriate to.
 * Completeness
 * Compactness
 * Connectedness
 * Continuous maps
 * Metric spaces
 * Contraction Mapping Principle

Comments by Msmithma

 * I have just submitted a major rewrite of the Sequences section. Amongst other things I changed the notation for sequences from $$\{x_n\}_{n=1}^\infty$$ to $$(x_n)_{n=1}^\infty$$.  This is more standard, easier to type, easier to read in source and will not conflict with the set theory book (if that is ever written).  It also inlines better (in the form $$(x_n)$$) when you have the "Recommended for modern browsers" math view option selected. J Bytheway 15:56, 26 Jan 2005 (UTC)


 * Incidentally, I would welcome discussion concerning the structure and approach that should be taken in this book, but in the mean time I shall continue to do things the way I think they should be done. J Bytheway 22:55, 28 Jan 2005 (UTC)


 * I too was a little overwhelmed by the work to be done in the set theory book, will start on Topology/Compactness tomorrow. Made minor edit to Bolzano-Weierstrass Theorem, Bolzano was spelt wrong and definition was not limited to the reals. Will check the proof tomorrow as I'm too tired for reasoned arguement now. Badge 01:31, 04 Feb 2005 (UTC)


 * I just did a major re-write/organization of "The real numbers" section. Added a couple sections on ordered sets and fields, didn't put much onto the field section.  I broke the main chapter of the section up into two pages so that the wiki stoped complaining about the size of the page and to make it easier to read.  Now the axioms are on one page and the properties/consequences are on the next. I also added a cover page as per the wikibooks style guide. Also add your names to the Real analysis/Authors  Page! Msmithma 07:42, 22 November 2006 (UTC)

--Msmithma 11:13, 4 December 2007 (UTC)
 * I just merged the edits on the natural numbers section with the original text, there was some redundant text that was removed and I'm not sure I did a great job placing the two new axiomatic approaches to the natural numbers but it restores the old flow while truing to incorporate the new material. We should probably choose if we want the natural numbers to start at 1 or 0, I vote 1.

Differentiation

 * Personally, I would like to see there be separate sections for Functions and Differentiation, rather than differentiation as a subsection of Functions as it is now. How does everyone else (assuming there is anyone else :-p) feel about this change? --Notanut 10:43, 6 Mar 2005 (UTC)

Topics Covered
Ch.0, Background 0.0 Sets (unless it's assumed the reader is already familiar) 0.1 The Real Numbers 0.2 Construction (though I still don't know what is planned for this) 0.3 Metric Spaces (maybe) 0.4 Regions (ie Neighborhoods, Open/Closed sets, Boundaries, Compactness... all that fun stuff) Ch.1, Real Variables and Functions 1.1 Functions 1.2 Limits 1.3 Continuity 1.4 Sequences 1.5 Series Ch.2, Differentiation 2.1 Derivatives 2.2 Stuff about derivatives 2.3 L'hospital (maybe) 2.4 Taylor's Theorem Ch.3, Integration 3.1 Riemann Integration 3.2 Fundamental Theorem of Calculus 3.3 Lebesgue Integration Ch.4, Advanced Topics 4.1 Measures 4.2 Fourier Series 4.3 Distributions 4.4 Hilbert Spaces 4.5 Other Forms of Integration (like Riemann-Stieltjes and so on) Obviously Ch.4's outline is wildly unorganized, and the rest could probably be improved upon. Suggestions are welcome, especially since I've never had formal training in Real Analysis. --Notanut 12:12, 8 Mar 2005 (UTC)
 * In response to the comment about structures and topics, I've started this thread. Personally, I don't like the current organization very much and I would also like to see a broader coverage of topics (ie a little basic topology - open/closed set, compactness and the like) in addition to coverage of topics currently deemed too advanced (especially integrals other than the Riemann integral... at least I think Lebesgue integrals are currently out because they're too advanced). I think a good organization would be something mor or less of the form:


 * In principle I agree with you. When I laid out the present structure I would have included more metric/topological stuff had it not been for the fact that there was already a Topology book.  I thought it would be better to have two seperate works, but heavily intertwined.  With luck, this would allow people who don't want to study topology the chance to learn real analysis independantly, and those who do the chance to learn the two together (which, I agree, is somewhat more elegant).  Writing it all out twice may aid learners who want a different presentation, but it nevertheless seems slightly silly when we have the opportunity to make good use of the wiki format to avoid it.
 * A presentation along the lines suggested above would certainly work, although I would rather put Lebesgue integration after measures, and relegate most of the other advanced topics to a book on functional analysis (with a link in the "where next?.." section of this book).
 * The construction section (about which you seem confused) is intended to describe ways of constructing the real numbers (out of the rational numbers, e.g. as equivalence classes of Cauchy sequences). When I learnt real analysis, this aspect was very much brushed over, and I feel it deserves more attention.  It's certainly not crucial for understanding of the other topics, however. J Bytheway 16:12, 3 Apr 2005 (UTC)


 * I provided a proof of the Squeeze Theorem, and added the equivelant characterizations of completeness that were promised in the "Real Numbers" section. If I did something terrible, let me know.   The proofs I gave could doubtlessly be improved, and I will keep editing them myself.  I did this more as a way of getting some text out there than anything else.  I'm a bit of a newcomer here, so I apologize for any faux pas I may have committed.  I think most of the changes I made are of a reversible nature, anyway.  Also, I had some problems with the math formatting that I couldn't fix.  If someone knows how to make things work right, please do.  L Culler 3:46, 11 May 2005

Set Theory Notation

 * I find some of the Set Theory notation odd. For example: $$\forall\epsilon>0:\exists N:\forall n \geq N:|x_n-a|\leq\epsilon$$,doesn't have an implication arrow.  Neither does any other theorem have an implication arrow.  I am used to seeing set theoretic statements written as such:$$(\forall\epsilon>0)(\exists N)(\forall n \geq N)\implies |x_n-a|\leq\epsilon$$, which is a little less ambiguous. But, $$(\forall\epsilon\in\mathbb{R}>0)(\exists N\in\mathbb{N})(\forall n\in\mathbb{N}\geq N)\implies |x_n-a|\leq\epsilon$$ is even less ambiguous.  What do you think?  --Fraxtal 22:44, 29 May 2005 (UTC)


 * Welcome to Wikibooks Math! MikeBorkowski 20:56, 11 May 2005 (UTC)

Why was the Real Analysis book moved be be a Topology sub-book?
Why was the Real Analysis book moved be be a Topology sub-book? Surely these should be separate books. Would anybody object if I moved Real Analysis to be its own book again? --JMRyan T E C20:11, 19 April 2006 (UTC)
 * I second that vote and will start the move --M. Smith-Martinez

Measure and Integration
I'm interested in begin a wikibook about measure and integration and i want to know if someone has some work done to cooperate or begin the book myself. --Bunder 05:13, 11 July 2006 (UTC)

Why Analysis?
As a newcomer I'm unsure if this belongs here, but one of the things which I struggled with (and which many other newcomers to analysis struggle with) is the question of why analysis needs to be done at all.

The common situation is that a first-year undergraduate mathematician will have studied calculus at A-level, concentrating on the how and trusting in the why (or seeing an intuitive, rather than rigorous, proof of the why). This is no bad thing IMO, but it does result in a feeling of confusion and disappointment as our undergraduates learn that they're going to spend their first year "learning differentiation and integration".

Of course, once the course is over it's more apparent why it was necessary, but until then the whole thing is rather bewildering.

So one thing I'd like to propose writing is a sort of streetmap of real analysis, explaining - in fairly untechnical language - how all the parts of the topic slot together, where everything fits, and why it's not the same as calculus.

Would this be a good place to add it? (I've yet to sign up for a username, but will likely do so soon...)


 * Ah... ignore that. I've just seen that the front page has exactly that. --81.187.181.66 14:53, 13 March 2007 (UTC)

Exponential Function
I am adding a chapter solely on the exponential and logarithmic functions. The usual way to construct these functions is to define the logarithm as $$\int_1^x\frac{dx}{x}$$ and $$e^x$$ as its inverse. Although the method I'm providing is longer, I believe it has its own merits.

I am also moving the section on rational exponentiation currently in the Continuity chapter to the new chapter. SPat (talk) 09:48, 12 April 2008 (UTC)

Cover
This is my temporary proposal for a cove page. Please feel free to comment or edit/replace as you may see fit

SPat (talk) 05:53, 24 May 2008 (UTC)

analysis book in portuguese
hi.. i'm writing the Real analysis's book in portuguese and if somepeaple wann to help in verification of demonstration, proof, ..., etc. Let welcome. Thank you!! Real analysis's book in portuguese
 * Thiago Marcel

Introduction
The introduction seems opiniated and vague. It might be best not to state that " the subject of Real Analysis may seem rather senseless and trivial " in the first paragraph. The introduction should rather explain why real analysis has an important place in higher mathematics, and how its study is necessary for the understanding of other more advanced or applied subjects. I also disagree that real analysis "is simply a nearly linear development of mathematical ideas you have come across". The subject is often presented in this way to students not majoring in math, but it is not true of the whole subjet. 216.246.250.81 (discuss) 13:44, 24 March 2018 (UTC)