Talk:Uniform norm

Notation suggests $L^{\infty }$ norm, which is not accurate

--171.65.37.29 (talk) 21:24, 14 April 2017 (UTC)

This article uses the same notation for the uniform norm of a function as the $L^{\infty }$ norm. While the two coincide for continuous functions, they are not the same for general measurable functions. That is, the uniform norm is $\max _{x}|f(x)|$, while the $L^{\infty }$ norm is the essential supremum, see Essential supremum and essential infimum. To see the difference, consider the Kronecker delta function, and see that the limit of its $L^{p}$ norms is not its uniform norm. It would be good to clarify that the uniform norm is not $\|f\|_{\infty }$ for more general functions.

Moved from "Talk:Maximum norm"

This seems to be a special case of the uniform norm. So, how about putting it all on one page?

Charles Matthews 17:44, 16 Sep 2004 (UTC)

I did not know there was a page on uniform norm. I created the article because I needed an explanation for maximum norm on the Chebyshev polynomial page. In any case I am a bit confused between the relation of uniform norm (is this the supremum norm?) and maximum norm. How is the maximum norm denoted $\|\cdot \|_{0}$ or $\|\cdot \|_{\infty }$. If your are any more knowledgeable then put it on the uniform norm page.MathMartin 21:30, 16 Sep 2004 (UTC)

I now added in a mention to maximum norm on uniform norm. Let me know if you're still confused about the relation between the supremum and maximum, because then I should explain it better. -- Jitse Niesen 14:11, 20 Sep 2004 (UTC)

I think that the term sup norm is quite frequently used for finite and infinite dimensional case. I just created that page as a redirect - since it means supremum norm, I made it point here, but this page is not very nice for beginners - while the page Chebyshev distance is not only a perfect stub, but also this name is less common (IMHO) and not so well chosen (at least if one wants to speak of the norm and not of the distance). — MFH:Talk 13:50, 23 March 2006 (UTC)

merge Chebyshev distance to Uniform norm

These both discuss the same concept, namely the d_∞ metric (aka, L^∞ norm), and in fact reference each other – shall we merge them?

Nils von Barth (nbarth) (talk) 07:54, 8 September 2008 (UTC)

My preference would be to merge the contents of uniform norm into Norm (mathematics), where it's presently not even mentioned, as well as parts as needed into Chebyshev distance. The problem with the present uniform norm article is that it jumps straight into dense abstract mathematics, whereas what most non-experts want to know about a Chebyshev distance is better described in that article. Dicklyon (talk) 21:08, 8 September 2008 (UTC)

Agreed on both counts:

• The uniform norm is mentioned in the section Norm (mathematics)#Infinity norm or maximum norm, but very briefly – it could use some expanding.
• A discussion of the uniform/Chebyshev norm should start with an elementary discussion, and the Chebyshev distance article does a much better job of this.

Regarding where to put the content, I’d think:

• put more details on Norm (mathematics)
• have one main uniform/Chebyshev norm article.

In my experience I’ve only heard it called the uniform norm, and that seems a more neutral term (and accords with uniform convergence, uniform continuity, uniform space), hence why I proposed the merge to uniform norm.

How does this sound?

Nils von Barth (nbarth) (talk) 02:20, 10 September 2008 (UTC)

Well, there's still the issue of distance versus norm, as well as Chebyshev versus uniform, for the main title. See also Euclidean distance, which doesn't even mention the relevant norm; maybe that should be fixed, too. But in a lot of fields, people are more used to thinking of distances, and don't bother to know about the norms that they are based on. Maybe we should see how these things are treated in some representative texts of various fields. Dicklyon (talk) 03:12, 10 September 2008 (UTC)

I’ve linked Euclidean distance to and from Euclidean norm.

Good point about “norm” vs. “metric”. (As per Relation of norms and metrics.)

Despite having more topology training than analysis, I’m inclined to call the article “norm” for 3 reasons:

1. The term “uniform norm” is more used than “uniform metric” (as per Google and books.google.com)
2. Calling it a norm is more information than a metric (it’s homogeneous and translation invariant)
3. The examples are almost all for vector spaces (indeed, the definitions on both pages are given only for vector spaces).

Nils von Barth (nbarth) (talk) 22:23, 10 September 2008 (UTC)

I'm still against the merge. Chebyshev distance is a well-known concept that deserves an article. A norm is not a distance, and there's no reason to try to do both in one article. Just as Euclidean distance has an article and Euclidean norm is covered elsewhere. Dicklyon (talk) 20:28, 21 September 2008 (UTC)

I formalized the alternative proposal here: Talk:Norm_(mathematics)#Proposal:_Merge_Uniform_norm_to_here. Dicklyon (talk) 20:34, 21 September 2008 (UTC)

Hi Dick,

As discussed at alternative proposal, I don’t think there’s much value in separating norm from distance: they’re very' close ($d(x,y)=|x-y|$; $|x|=d(x,0)$) and anyone interested in one is likely interested in the other (unlike the elementary Euclidean case, which is of wide interest, hence it’s worth avoiding discussing a norm).

I don’t feel as strongly about finite dimensional versus infinite dimensional (function space) – they are close, and I think they work well together, but I’m not averse to having separate articles.

Nils von Barth (nbarth) (talk) 00:24, 22 September 2008 (UTC)

That's where we disagree. Lots of people who are not mathematicians care about distances, and never heard of a norm. That's why the two articles have so little in common. Dicklyon (talk) 00:26, 22 September 2008 (UTC)

The articles differ because uniform norm discusses functions, and Chebyshev distance discusses finite dimensions, not because of norm versus distance.

Also, Chebyshev implicitly discusses norm when it says a "circle of radius r".

Agreed that non-mathematicians care about distances, but who cares about the Chebyshev distance and would be thrown for a loop by "norm"? I’ve only seen it in the context of Lp norms.

Nils von Barth (nbarth) (talk) 02:49, 22 September 2008 (UTC)

Nils, you should be looking to convince someone else, as you're not going to convince me. Don't do the merge without at least some support for the idea. Dicklyon (talk) 03:56, 22 September 2008 (UTC)

• Don't merge, per arguments by Dicklyon. Also, I've noticed that making single, large articles that try to "say everything" are daunting for the reader (lots and lots to read, which can be discouraging), and are difficult for editors to maintain (since large articles get frequent revisions, and as a result, little parts of the article get chipped away, and are lost, because no one notices, because they're hidden by the large changes.). Smaller articles are more easily monitored for deleterious changes. linas (talk) 01:20, 9 November 2008 (UTC)

Also, norms are not 'almost the same as' distances. Metrics have to satisfy the triangle inequality, there are plenty of norms that fail do so, and thus cannot be used to define a metric. These are not interchangeable concepts. Silly me, what was I thinking of? Something else, clearly linas (talk) 01:25, 9 November 2008 (UTC)

Given opposition, I’ve not merged them.

I’ve added {{otheruses4}} though as hatnote so that they prominently reference each other, in case someone looking for one finds the other.

—Nils von Barth (nbarth) (talk) 20:59, 16 May 2009 (UTC)

Please do not assume because it makes an ... of you and me. Examples, illustrations and physical meaning/interpretation.

First, the best thing we can do is to not assume the reader nows much. Therefore, we should define or state everything in the definition of sup norm leaving no susceptibility to confusion, such as that |f(x)| in the curly brackets means the absolute value of f(x) which one could possibly, though quite unlikely, confuse to be the norm of f(x) so then the sup norm to them would mean a norm of the norm of f(x).

Also, after the mathematical definition of the sup norm is given, I think it should be pursued by an explanation explaining exactly what the sup norm is (or does). For example one could say, the sup norm, "it first computes all the f(x)'s and takes their absolute values, then the highest absolute value of f(x) namely |f(x)| is taken to be the sup norm." One could also add that the step of taking the highest absolute value of f(x) is incurred by the function sup{ } and then add a link to "sup{ }".

Finally, I also find that there should be few different examples along with a few geometrical illustrations of sup norms in different spaces to engender others to have a more visual understanding of the sup norm, so that they are not left with simply an abstract notion of the sup norm, and thus that it may then, dawn upon them, that the sup norm is one of the most easiest things ever.--Gustav Ulsh Iler (talk) 01:04, 21 October 2009 (UTC)

Applications to Integer Programming

I have heard that the infinity norm has applications in integer programming (which currently has a very poor Wikipedia article). Maybe this application can be discussed in this article. Bender2k14 (talk) 02:55, 29 August 2010 (UTC)

Proof

Given that $||x||_{p}=(|x_{1}|^{p}+\dots +|x_{n}|)^{1/p}$, where can one find a proof that the infinity norm is, in fact, the maximum of the absolute values? That is, I'd like to understand how $\lim _{p\to \infty }(|x_{1}|^{p}+\dots +|x_{n}|)^{1/p}=\max\{|x_{1}|,\dots ,|x_{n}|\}$. Rudin does not get into that in the page mentioned in the references. Surement (talk) 02:13, 26 December 2012 (UTC)