Talk:Banach fixed-point theorem

Lipschitz constant

The smallest such value of q is sometimes called the Lipschitz constant.

Any such q is a Lipschitz constant. There isn't the Lipschitz constant. 84.191.234.177 (talk) 20:09, 26 June 2008 (UTC)

Contractive Mapping Theorem

Maybe a little note about Edelstein's contractive mapping theorem, a generalized version of the Contraction mapping theorem, which sets Lipschitz constant to be equal to 1 but makes the inequality strict should be added? 202.36.179.66 (talk) 02:06, 24 September 2008 (UTC)R

With what additional condition? Compactness? Algebraist 12:49, 24 September 2008 (UTC)

Yep, compactness is the extra condition required to make the result go through. It was proved by M.Edelstein in the 60s and a constructive version of it was proved by D.Bridges in the 90s. The constructive formulation is, as one would expect, not nearly as general. Constructively, the proof for the existence of the fixed points is proved for all bounded euclidean spaces but the convergence of the iterates is only proven for up to 2 dimensions.121.73.122.128 (talk) 10:09, 26 September 2008 (UTC)R.

Proofs are inappropriate for Wikipedia

Proofs are not inappropriate for Wikipedia. However, the proof presented here is unnecessarily long and I think it was made that way so someone could plug their silly new proof. Someone with time on their hands should cut out the extra garbage. Sincerely, Zach.

Hear, hear.80.47.199.50 (talk) 12:17, 9 January 2016 (UTC)

Did my best.Student298 (talk) 21:38, 16 November 2018 (UTC)

Wikipedia is not a textbook, so proofs are generally not appropriate in articles. Even when a proof is of special interest in itself, e.g. for historical reasons (which is not the case here), it suffices to outline the main ideas and provide a reference for the deails. So I propose that the "Proof" section be removed. There may be a place for it in Wikibooks, hopefully. All the best, --Jorge Stolfi (talk) 02:03, 30 December 2009 (UTC)

Since you put it as a general principle, let me take the opportunity for some general remarks too. An encyclopedia like wikipedia provides information at various levels, and this is agreed as one main reason of its success. As a consequence, we must live with the idea that wikipedia should also contain some information which is of very high interest to very few users. The alternative entails a strong risk of devolving quickly into a flat list of trivialities. Here the issue of respecting the needs of a minority is not just in order to aknowledge an instance of democracy: if we remove the specialized material, most of the specialized contributors will eventually leave, and the general, non-specialized information will suffer as well. In particular, in many cases, a sketch of a proof is a very valuable quick reference, even if it is only available, and of interest, to the mathematically educated people. That is, to a small minority: though vital to wikipedia. In the present particular case, however, I agree that the proof is quite a bit longer than needed, and the main issues of the result may be stated a bit better. --pma (talk) 11:29, 30 December 2009 (UTC)

I believe that we mostly agree. I have nothing against specialized material per se, and indeed I believe that there is no such thing as a topic that is "too specialized" for Wikipedia. I also believe that there are mny cases where a sketch of a proof does no harm. However, I also believe that each article must be organized so that its is most convenient to read for most readers; and I do not believe that readers will be interested in proofs. These are useful only for two things: in books and journals, to prove to other skeptical fellow mathematicians that a claim is true; and in textbooks, to teach students to "think mathematically". The former use is not applicable here since Wikipedia does not accept original research. (And, by the way, a proof specifically written for Wikipedia is uncomfortably close to being original research itself.) As for the second use, Wikipedia is not (and should not be) a substitute for textbooks; and students already have plenty of proofs in their textbooks and classes. So, for most Wikipedia readers, proofs will be just useless clutter standing in the way of the facts.
Look at articles on other sciences: there you will find lots of factual statements but hardly a hint of the detailed experiments and statistical analyses that were used to prove them. Except for experiments of great historical value, the reader is expected to look for such details in the cited sources. Why should math articles be different?
However, I will not fight on this point; I only ask you to please consider whether the article gains or loses by including a proof of the theorem. All the best, --Jorge Stolfi (talk) 18:44, 30 December 2009 (UTC)

Proofs are not simply meant to validate claims like statements of $p$ values and statistical methods. Proofs are different beasts that communicate additional information about the results. A $p$ value in a journal does not aid the reader in understanding; however, a proof does. Likewise, it is not unprecedented for encyclopedia articles to describe how experiments are performed (e.g., the oil drop experiment). In the cases where a proof is destroying the readability of an article, it can be put inside a collapsible box. See Bernoulli's principle#Derivations_of_Bernoulli_equation for an example. —TedPavlic (talk/contrib/@) 22:25, 30 December 2009 (UTC)

I disagree that proofs should be completely removed from an article with reference to a general guideline as opposed to specific reference to the article in question. Although an article on compact spaces should not contain a proof of the Gelfand-Naimark theorem (for instance), the article on the Gelfand-Naimark theorem should contain such a proof. In fact, an article on "theorem X" should contain a proof of "theorem X"; after all, this is an encyclopedia and we should strive to include the most important details about "theorem X". What is more important than its hypothesis, implications within mathematics, and its proof? Applying general guidelines such as "mathematics articles should not contain proofs" should be done with care, especially if the article in question is about a proof. --PST 13:20, 3 January 2010 (UTC)

Applications

The contraction mapping theorem can be used to prove the existence and uniqueness of a solution to a differential equation. Can it be used for a stochastic differential equation? Jackzhp (talk) 18:46, 11 July 2010 (UTC)

Journal of Fixed Point Theory and its Application

...is not what this journal is called!80.47.199.50 (talk) 12:14, 9 January 2016 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Banach fixed-point theorem/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

--Cronholm144 20:44, 21 May 2007 (UTC) I am new in Wikipedia, but I think it is possible to improve this article. For example, there are a lot of applications of the Banach fixed point. I want to do this list. --Incredulo (talk) 00:27, 24 June 2008 (UTC)Incredulo

Substituted at 18:09, 17 July 2016 (UTC)