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Why is the length of a vector called a norm? And who was the first person to use this word in this meaning? In ordinary language, "normal" means something usual without any special property. In geometry, however, normal means something special, namely perpendicular (or orthogonal). This is a surprise for me. And why do we call the length of a vector "a norm of the vector"?
Pokyrek ( talk) 07:59, 26 February 2019 (UTC)
Does anyone know what the modulus of an element of an arbitrary field is? Does the notion of modulus exist in every field?
It is not obvious for me why we have the implication that p is positive, in none of my references (or the book cited) is the norm defined as anything but positive. Could you please elaborate, why does it follow that it is positive through this definition? See the article on normed vector space for example.
I don't know what the || notation means. From context, I think I get that it is analagous to absolute value or cardinality, but for vectors. Will someone please add something explictly introducing the || notation either here or on it's own page?
Kal Culus
I do not understand your question. || is a function on a vector space. We write |.| to indicate the function has to be feed a vector. We could also write f(x), with f the norm and x an element of the vector space V. The properties of the norm function are given in the article. I do not know how to make this more explicit in the article. MathMartin 13:33, 29 Oct 2004 (UTC)
I will try to answer your question "I don't know what the || notation means." by an example. If a=(3,4) then ||a||=. Hope this helps.
Pokyrek ( talk) 07:38, 26 February 2019 (UTC)
Would it be agreeable to merge the material from here back into normed vector space? I can see a couple of reasons for this:
AxelBoldt 19:59, 22 Apr 2005 (UTC)
I did not separate norm (mathematics) and normed vector space but I did separate metric (mathematics) and metric space recently and I think most of the arguments are valid for those pages as well. See Talk:Metric_space#Split_metric_space_into_metric_.28mathematics.29_and_metric_space
MathMartin 20:53, 22 Apr 2005 (UTC)
If you are not convinced by my arguments feel free to merge the pages again. MathMartin 13:52, 23 Apr 2005 (UTC)
This article defines norms for any field F, but there is a problem with axiom 2 since it relies on a real-valued function |a| defined in the field. What does that mean for a finite field, for example? -- Zero 06:28, 23 Apr 2005 (UTC)
I agree something does not make sense. If F=C are the complex numbers, then axiom 2 has no meaning, as C is not an ordered field, so we cant say that p(u+v) <= p(u) + p(v) — Preceding unsigned comment added by 186.18.76.220 ( talk) 01:53, 28 September 2011 (UTC)
On the Gauge disambiguation page, I tried to come up with a short one-line description, and wrote:
"* Gauge (mathematics), a semi norm, a concept related to convex sets.
Unfortunately, I don't understand this stuff well enough to know if I butchered the description or not. Could somebody please check it out and correct if needed? Thanks. RoySmith 8 July 2005 11:58 (UTC)
Somehow I seem to remember there being a person's name associated with the process of turning a seminorm into a norm by modding out by the vectors of norm zero, in just the same way that you turn the space of Lebesgue square integrable functions into a Banach space by identifying functions that agree almost everywhere. I came to wikipedia to look up this name, but couldn't find it. Am I imagining this? - Lethe | Talk 02:46, July 11, 2005 (UTC)
In my Finite Element Analysis course, I keep running into the L2 norm,
and the "energy norm" (which apparently, for most solid mechanics applications, denotes energy):
Can someone who knows more than I on the subject add information on these? — BenFrantzDale 03:49, 27 September 2005 (UTC)
We need to generalize max to sup for infinity norm. MathStatWoman 09:26, 22 January 2006 (UTC)
This article has been very helpful, but I don't understand this about the infinity norm: "The set of vectors whose ∞-norm is a given constant forms the surface of a hypercube of dimension equivalent to that of the norm minus 1." I understand why it form a hypercube, but what is the part about the dimension? Oh never mind, I get it now. I just didn't know a norm had a dimension and as far as I can tell that is never mentioned in the article. I thought of the dimension of the hypercube as the size or edge length or something like that. Which I guess would be equal to two times the norm. 150.135.159.218 ( talk) 18:47, 15 October 2008 (UTC)
I see this portion as inconsistent since : Let p ≥ 1 be a real number. ...and for p = we get the infinity norm -- Kmatzen ( talk) 15:10, 21 October 2011 (UTC)
If we have an article entitled "Norm(mathematics)", should we not include the concepts of Banach spaces, norms based on the general Lebesgue integrals as well as countable sums, and sup norms? To quote from Banach spaces: "The Banach space l∞ consists of all bounded sequences of elements in K; the norm of such a sequence is defined to be the supremum of the absolute values of the sequence's members." MathStatWoman 13:04, 22 January 2006 (UTC)
Someone should add some links (in this article) to other articles explaining norms, and the sup norm as an example, since it is so important in other areas of mathematics. Is that all right with everyone out there? MathStatWoman 18:33, 23 January 2006 (UTC)
I found a weak norm being used at http://www.sm.luth.se/~johanb/applmath/chap3en/part3.htm. Should this be in the main article? -- Sunnyside 10:40, 9 February 2006 (UTC)
I prefer to define a semi-norm as in the latest version just posted today. Positive homogeneity and the triangle inequality are the only two properties you need, and it seems silly to keep carrying around positive definiteness when it follows. Perhaps I am missing something as I have not seen this anywhere. Jenny Harrison 12 July 2006
Hi, on a possibly related note, I noticed this under Notes:
A useful consequence of the norm axioms is the inequality ||u ± v|| ≥ | ||u|| − ||v|| | for all u and v ∈ K.
Would it be okay to move this along with positive definiteness as another property that follows from the definitions? The Absolute value article reads much better, witha section of fundamental properties and another for derived ones. MisterSheik 03:03, 17 July 2006 (UTC)
The terminology "positive homogeneity" is misleading, especially since the article then links to a different and much more common definition of 'positive homogeneity'. This property should be called 'symmetry' or possibly 'symmetric homogeneity'. Cerberus ( talk) 13:15, 26 July 2013 (UTC)
References
Perhaps there is a L_(-infinity), with a unit circle of a cross (0 width)? —The preceding unsigned comment was added by 67.183.154.231 ( talk) 04:01, 5 December 2006 (UTC).
Resolved |
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Regarding edits. I didn't confuse hamming distance with the taxicab norm. My interpretation is not incorrect, hamming distance and 1-norm are equivalent reductions(HAMMING ≤ 1-norm). You could consider the hamming distance to be the 1-norm of the binary vertex representing the difference between two vertices on a Hypercube graph. (hamming(A, B) = "1-norm"(A-B) or "1-norm"(A ⊕B)). In addition Hamming_distance states "and the Hamming distance of the strings is equivalent to the Manhattan_distance between the vertices." and you can see that "Manhattan distance" is redirected to Taxicab_geometry. These two concepts are very similar and indeed they are reducibly equivalent. -- ANONYMOUS COWARD0xC0DE 10:01, 3 January 2007 (UTC)
In the part where it talks about how every seminormed vector space induces a normed vector space V/W, I don't understand the formula (||W+v|| = p(v)) that is given for the induced norm on V/W. For one thing, what does it means to add the vector space W to the vector v?
There are a lot of names for various norms. We need some cleanup to clarify this. There's "Euclidean norm", L2, taxicab, etc. It should be made clear in the introduction which of these is which and some of the redirects need to be looked into. For example L1 norm redirects neither here nor to Lp space. —Ben FrantzDale 00:53, 11 October 2007 (UTC)
To add to this cleanup request, I find a paragraph in the discussion on Hamming distances which refers to B-norm and F-norm without any hints as to what they are.
161.84.227.13 (
talk) 13:29, 16 September 2014 (UTC)
The L1 circle is described as a rhomboid when it is a square. Any objections to fixing this? -- Vaughan Pratt ( talk) 19:22, 26 June 2008 (UTC)
Just as Euclidean distance has an article and Euclidean norm redirects to a section in Norm (mathematics), so should Chebyshev distance continue to have an article and Uniform norm redirect to a section here that's prsently missing. Let's merge it in. This is an alternative to the proposal to merge Chebyshev distance into Uniform norm, which seems backwards to me. Dicklyon ( talk) 20:33, 21 September 2008 (UTC)
Note that in some contexts, the term "semi" is used when the triangle equality doesn't hold. In some cases, it is used synonym to "pseudo" (non-zero vectors of length 0 allowed). However these don't necessarily fall together (at least not for distance functions), do they? E.g. euclidean-like distance in a torus space (using the shortest path on the torus) doesn't fulfill the triangle equality, but the condition d(x,x)==0 <=> x=x holds. Unfortunately these terms are totally mixed up in all the related articles here on wikipedia. 138.246.7.155 ( talk) 16:35, 24 November 2008 (UTC)
I would like to second this. The current article other wikipedia articles are not clear on pseudonorms and seminorms. 67.214.224.120 ( talk) 15:58, 25 March 2016 (UTC)
The article defines the p-norm of a vector x in the usual way -- as the pth root of the sum of the pth powers of its components.
Two lines later it states, referring to this definition:
"This formula is also valid for 0 < p < 1, but the resulting function does not define a norm, because it violates the triangle inequality."
Agreed that it violates the triangle inequality. But I do not understand what "the formula is also valid" means here. Simply that the formula can be evaluated, i.e., that it makes mathematical sense? In that case, why exclude negative values of p ? (And why not also mention what happens when p -> 0 and when p -> -oo ?) Daqu ( talk) 06:55, 26 December 2008 (UTC)
I could not find an article or any mention of the following norm,
where A is positive definite matrix, and () is the inner product. Does this norm have a name in English? ( Igny ( talk) 15:38, 18 January 2009 (UTC))
To add to the overall clarification in the examples section, it might be a good idea to add some numerical examples in with all of the symbolic examples. —Preceding unsigned comment added by 132.235.19.87 ( talk) 13:03, 28 April 2009 (UTC)
regarding this in the introduction: The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude. Isn't the "length" of the arrow determined by the norm? Is there anything really more fundamental about "length"? The statement seems circular. 76.175.72.51 ( talk) 16:26, 20 September 2009 (UTC)
I find it instructive to link to related concepts that answer the question "What if you remove axioms?". In particular, I am curious about what happens when you drop axiom 1 from the definition of norm? Do you get a function that defines a metric? I think so, but am not sure...
Metric requires:
Norm requires:
The thing that throws me is the note at the end of the "norm" axiom. It seems that criterion 3 is prescribing positive-definiteness. But the note says positivity and non-degeneracy (in the sense that ) is implied by the first two "norm" axioms. If so, then all axiom 3 adds is definiteness. Is that right?
If all of axiom 3 is required, then dropping axiom 1 seems to leave you with a function, p, still defining a metric, . However, if criterion 3 is really a note, implied by axioms 1 and 2, but not an axiom defining "norm", then criterion 3 would fail if we remove axiom 1 and so "a norm without criterion 1" wouldn't define a metric without criterion 3.
Why is axiom 3 so long if much of it is implied by axioms 1 and 2?
Which axioms can be removed from "norm" to still get something that defines a metric? —Ben FrantzDale ( talk) 15:13, 28 January 2010 (UTC)
One of the key properties of norms is that they are all equivalent up to constant factors. In particular, if and are two norms, then there exist some positive constants C1 and C2 such that:
for all x in a vector space (for the same constants). As a consequence, it doesn't matter what norm you pick if you don't care about constant factors, which is very convenient for proving lots of asymptotic bounds, talking about stability of algorithms, etcetera. This is discussed e.g. in Trefethen and Bau, Numerical Linear Algebra.
(At least, this is true for finite-dimensional vector spaces; I'm not sure about the infinite-dimensional case.)
(Proof in brief: prove that any norm is continuous, reduce to considering the unit ball by dividing by , then invoke fact that continuous functions achieve their maxima and minima on a compact set to obtain C1 and C2.)
Anyway, it seems like this should be mentioned somewhere.
— Steven G. Johnson ( talk) 02:58, 15 March 2010 (UTC)
It's possibly a simple oversight, but complex numbers all have norms, or at least that was what Euler called for the complex number x + iy. And yet this article has no section on complex number norms. The complex number wiki has a section on how to compute this value for complex numbers, but doesn't CALL it a "norm," but rather modulus, absolute value, or magnitude. But if a complex number is considered a vector, then obviously its magnitude is the norm. Has the Euler usage entirely gone by the wayside?
It's not obvious how to compute a complex norm, as you multiply by the complex conjugate, which means you essentially disregard the quantity i. If you didn't, and included it, you'd get a different kind of norm (the Minkowski norm). Is there something in the terminology here that I'm missing? Obviously I'm no mathematician. S B H arris 01:55, 8 April 2010 (UTC)
Regarding this edit [1] I don't believe 0 is an imaginary number, so the imaginary numbers are not an abelian group. much less a vector space. — Carl ( CBM · talk) 10:44, 27 October 2010 (UTC)
A moment ago I wrote in an edit summary
And after clicking, of course, I noticed that I meant (n+1)- ball. Oh well. — Tamfang ( talk) 21:25, 11 July 2011 (UTC)
User:Rheyik recently added:
Uh, what? I guess these definitions concern something fancier than the plain old vectors that appear to be the context of the rest of the article. — Tamfang ( talk) 01:20, 12 December 2011 (UTC)
Why does Pseudonorm redirect to this page, when the page doesn't mention "Pseudonorm" anywhere? I have seen pseudonorm used in the sense of the Hamming-distance from 0, but there maybe other conventions. It would be nice to mention it, even if only to say that there is no consensus on the terminology. Lavaka ( talk) 14:35, 20 April 2012 (UTC)
I improved the Notation section and put it after Definition section which is a more logical place for it.
I removed the claim that set cardinality is a norm and should be noted by double vertical bars because I cannot see how sets would make up a normed vector space (if there is a natural way to do it, it would be interesting to know) and single-bar notation is also much more common for cardinality.
I removed the claim that denoting norm by single bars is generally discouraged. Actually, the the only case where I have encountered the single-bar notation is the case of Euclidean spaces, and for them, such usage is widespread, so a statement that it has been discouraged should be backed up by an authoritative source. I also removed the suggestion of denoting matrix norm with double vertical line since matrix spaces are special cases of vector spaces and I haven't seen their norms being denoted otherwise. (I personally do not see a problem with norm being confused with absolute value in case of Euclidean spaces, since the Euclidean norm in 1-dimensional space is exactly the absolute value. For matrices, I would rather blame the notation of determinant with single bars as a source of confusion, since for the usual matrix norms, the norm of a 1-by-1 matrix is the absolute value of its only element but it may differ from the determinant of the matrix by sign.)
If someone knows some sources where norms on generic vector spaces or matrix spaces or some other specific spaces are denoted by single bars and can find a source that such notation is discouraged, then such suggestions can be added back to the Notation section. Jaan Vajakas ( talk) 16:30, 14 September 2012 (UTC)
I removed the claim "Although every vector space is seminormed (e.g., with the trivial seminorm in the Examples section below), it may not be normed." since it is not true (for vector spaces over subfields of ). Indeed, for a field F, algebraic vector spaces over F are characterized by the cardinalities of their Hamel bases. For a vector space Hamel basis , , where A is an arbitrary (possibly infinite) set, we can consider any p-norm w. r. t. the basis, e. g. the 1-norm . Jaan Vajakas ( talk) 17:07, 14 September 2012 (UTC)
It wasn't my idea to include it...
But the more formal equation is:
— Arthur Rubin (talk) 21:34, 7 January 2013 (UTC)
To editor Quondum: an example of the confusion of symbols is the ‖Xegwi language redirect, which I just worked with today and which prompted me to mention the confusion here in this article. – Paine 16:57, 9 May 2015 (UTC)
I removed references to a "B-norm" (which as far as I can tell just means "norm") and attempted to clarify the meaning of "F-norm" in this article. The definition, from Rolewicz's book is the one I gave in the article, i.e. it is just where d is a distance in an F-space. That's not very satisfying and Rolewicz also gives some intrinsic conditions, namely is an "F-norm" if: iff x=0; ; and the triangle inequality holds.
I didn't feel like writing these out in the article, largely because I am unsure that this F-norm business belongs in here. Maybe it should go onto the F-space article, but I'm not really qualified to work out whether it is significant or interesting.
While tidying I also tried to make the language a bit less judgmental. I don't know what "some engineers" means - I think that some mathematicians also call things norms when they aren't norms. Anyway. 130.88.123.107 ( talk) 14:20, 24 November 2015 (UTC)
The article doesn't mention the process of normalization, although it would seem to be relevant. 75.139.254.117 ( talk) 05:15, 28 December 2016 (UTC)
A discussion is taking place to address the redirect L2 norm. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 June 11#L2 norm until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 ( talk) 11:38, 11 June 2020 (UTC)
A discussion is taking place to address the redirect L2 norm. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 June 24#L2 norm until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 ( talk) 14:50, 24 June 2020 (UTC)
Mgkrupa added recently two new sections to this article, "Basic properties", and "Normability". For the reasons that follow, these sections do not belong to this article. By lazyness, I have reverted the article to the last version before this addition, and this resulted in reverting also edits (including by myself) to the other parts of the article. Mgkrupa restored my edits and the reverted sections. I'll remove these sections again for the following reasons.
Section "Normability" is not about the subject of this article, not even about normed vector spaces, it is about a property of topological vector space. It has nothing to do here.
Section "Basic properties" is not about basic properties of norms as suggested by its title. It starts by a nonsensical sentence implying the concept of a ball of given radius in a vector space that is not supposed to be a normed vector space. If this is fixed by supposing that the vector space is normed, most properties are not basic properties of norms, but properties of seminorms defined on a TVS or a normed vector space. Only properties 1, 3, and 4 are basic properties, but they are stated as properties of seminorms, without stating that they are also true for norms. All other properties are properties of seminorms on a topological vector space. This does not belong the first section after the definition. D.Lazard ( talk) 08:39, 16 July 2020 (UTC)
In the Equivalence section, where is says "For instance, if p > r > 1", Shouldn't it actually say "p > r >= 1"? — Preceding unsigned comment added by 2601:644:400:5FF0:5E0:73E:C912:D92B ( talk) 21:37, 25 November 2020 (UTC)
In many articles when the field in question is ℝ or ℂ, it is written . In this article it is 𝔽 or 𝔽. I'd like to prefer the first – and only upright. – Nomen4Omen ( talk) 21:56, 13 December 2020 (UTC)
@ D.Lazard: Hi, the word "non-negative" (with a hyphen) is more common than nonnegative. For example see /info/en/?search=Sign_(mathematics)#non-negative_and_non-positive. Hooman Mallahzadeh ( talk) 17:01, 8 March 2022 (UTC)
An editor has identified a potential problem with the redirect L² norm and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 May 4#L² norm until a consensus is reached, and readers of this page are welcome to contribute to the discussion. fgnievinski ( talk) 22:03, 4 May 2022 (UTC)
An editor has identified a potential problem with the redirect L2 norm and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 May 4#L2 norm until a consensus is reached, and readers of this page are welcome to contribute to the discussion. fgnievinski ( talk) 22:04, 4 May 2022 (UTC)
An editor has identified a potential problem with the redirect Square norm and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 May 4#Square norm until a consensus is reached, and readers of this page are welcome to contribute to the discussion. fgnievinski ( talk) 22:04, 4 May 2022 (UTC)
An editor has identified a potential problem with the redirect Pseudonorm and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 May 11#Pseudonorm until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer 1234qwer 4 20:35, 11 May 2022 (UTC)
In the 'Euclidean Norm' section, the Euclidean norm is first denoted , then as . Is the subscript 2 important in the first instance? If so, why does it disappear? And if not, why include it in the first place? The article should either be consistent, or explain why there is a difference between the two notations. Lhommedacier ( talk) 09:51, 8 August 2023 (UTC)
We use p in
Given a vector space over a subfield of the complex numbers a norm on is a real-valued function with the following properties, ...
and in the section on p-norms
Let be a real number. The -norm (also called -norm) of vector is ...
I think that using p for both these roles is confusing and I would like to change the former. (The latter is very much conventional use.) Is there a letter other than p that is typically used in textbooks for a function on a vector space that is a norm? — Quantling ( talk | contribs) 14:32, 6 September 2023 (UTC)
The article claims that some people define a pseudonorm by weakening the identity
to an inequality:
However, the claim needs more substantiation—the setup described by the reference (Kurosh's Lectures in General Algebra, 1965, via Encyclopedia of Math) is different; Kurosh talks only about normed rings, where belong to some common ring . Does anyone have a better reference? Thatsme314 ( talk) 22:17, 22 April 2024 (UTC)