Talk:Norm (mathematics)

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confusion, and out of context they are just symbols. You have not given any Unicode codepoint for the lateral click either (as has been given for the parallel symbol, and which is evidently U+x01C1), so I can't figure out the disambiguation intent. The disambiguation of Unicode symbols itself is moot anyhow, since Unicode does not assign semantics to symbols; it only provides the symbol, its codepoint, a general categorization and a brief name. It does not prescribe when or in what contexts they should be used. In fact, the alternative description appears to be "double pipe", and in this sense it would be perfectly usable to indicate the norm. I'm not particularly trying to go against what you want to do, it is just that I can't see the point/value of the addition. —

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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.)

95: 85: 64: 31: 2125:. 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 but important 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. 4276: 4259: 4693: 4644:

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.

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most don't realize it is not needed. It becomes a mantra that may be comforting but is pointless to carry it around. The lean definition actually reads very well, for it looks a lot like the definition of a linear transformation, a kind of sublinear functional, as it were. So I am going to try again to make this read well for you.

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mentioned, though "positivity" (as you people said) could be omitted. More over, I didnot understand at all why you people are thinking of "linear transformation" here!. my word is simply that we need all 3 properties.. positive definiteness, subadditivity and homogenity, compulsarily but not positivity to define NORM.

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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

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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 -:

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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

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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

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applies. However, consistency is more important. Later, in the article, there are (in the same section) "non-negativity" and "nonnegativity". So this must be changed, and we must choose between the two spelling. I agree that "non-negative" is more common in Knowledge. So, I agree to add an hyphen to

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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,

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Leave the discussion of which definition to be followed officially. First of all, Did you people check completely the if and only if condition?? I think you are talking only one way i.e. "||v||=0 if v=0" , but not "||v||=0 only if v=0". which makes the condition of "positive definiteness" must to be

3538:

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

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I would have done it already if this discussion hadn't taken place, but since it has, I'll put it up for comment here first. In the definition, I propose to replace "3. p(v) = 0 if and only if v is the zero vector (positive definiteness)" with "3. if p(v) = 0 then v is the zero vector (separates

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Let's think about this again. When we define a linear transformation T of a vector space, we do not usually list the fact that T(0) = 0 in the definition. We would appear ignorant if we did so. Similary, there is no need to include positive definitness. Since everyone has memorized it this way,

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I really don't know whether it's possible to make the definition of norm general enough for more fields. I guess for the norm to exist, we need some kind of map from our base field to the reals which will itself satisfy the axioms of a norm. a sort of subadditive field homomorphism to R. Do such

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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

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I still like the the way "absolute value" reads, and think that this article could benefit from immitating that model. My view is that all of the derived properties worth mentioning should be mentioned together. Also, I think that the positive definiteness is not that hard to derive :) (from 1,

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you're right, on the hypercube the hamming distance and the l1 distance are the same. however, the rest of the article is using vector spaces over the reals as examples. i think introducing the hamming distance at that point in the article confuses the reader. perhaps you could make a breakout

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As MisterSheik points out, the derivation of positivity is rather trivial. The adjective "crucial" is subjective. All properties are crucial when you need them. The question comes down to this: what is fundamental and what is derived? If a derived property is included in a definition, then it

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I see your point. However, T(0)=0 is trival to derive and not that essential for linear transformations. For seminorms, the positivity is crucial, and is important enough to actually state it as a definition. It is also rather clumsy to derive, and people better spend their attention on something

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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

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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

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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.

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must be verified each time one needs to check that a definition holds. It might be helpful to compare the definition of a linear transformation with the definition of a seminorm, with and without positivity. To show a linear transformation T is linear, we do not have to verify that T(0) = 0.

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is a title that should begin with the lateral clicks symbol, but instead it was created with the norm symbol leading it off. Someone caught this and moved the page to a title that uses the correct symbol. This is a case where the norm symbol was used in error because it was confused with the

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I fail to see what confusion you are referring to. To think that someone might see the symbol for a lateral click, and say to themselves "oh, that maybe means the norm as used in mathematics, let's look that up" seems a little far-fetched. Context distinction would be enough to eliminate any

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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.

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Okay, I see you're coming at it from the direction of "Here is this symbol, what does it represent?" – there may even be wikilinks that confiuse them. From that perspective I guess that it makes some sense. Ideally, each symbol should direct to an article about the symbol (or a category of

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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?

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Unless your "Oh never mind" is something I'm missing, the wording is ambiguous. A norm is always a scalar, so if you take "dimension" in the number-of-components sense, then the phrasing is wrong. Taking the other meaning -- edge length -- the article is just wrong. I just rephrased it.

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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.

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As for whether the finite and infinite case should be separate articles or merged into one article, I have no strong feelings – I think they’re very close and should go together, but one can reasonably argue that the R case and the function space case are conceptually pretty

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I guess one needs to have a section in this article discussing norms on function spaces. That seems to be missing now. Then, that norm would be most welcome, but I never heard it being called weak norm though, some references for that (except the web page) would be needed.

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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.

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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

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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.

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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.

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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'.

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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?

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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

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modulo the subspace of vectors of zero seminorm, on which the (induced) seminorm (the seminorm is independent of the representative of a class) is a norm. (But I don't know whose name could be associated to this -- maybe you think of the

5085:. 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. 2932:. 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. 3797: 3123: 2131:

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.

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W + v denotes an element of the quotient space V/W; it's the equivalence class . say we're in R^3 and W is a 2-d subspace, then W + v is the affine plane obtained by translating W by v. we should also check that the definition

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It is trying to be friendly to readers who already know what "length" means but came here to find out what "norm" means. So the point is "you already know one norm under the name 'length', it is called the Euclidean norm".

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Ummm, this section is a discussion from two years ago. If you want to merge the articles, I think you need to make a fresh argument. I think you also need to make a sound refutation of MathMartin's arguments above.

1452:||u ± v|| ≥ | ||u|| − ||v|| | is indeed a derived property and followw very easily from the triangle inequality. Positivity is much more crucial and not as easy to derive and I think it should stay in the definition. 792: 5522: 1293:

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?

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I'd support this merge. Neither definition can be made without making both and neither can be understood without reference to the other. If we want to separate out examples the articles should be something like

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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

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I don’t think separating “norm” and “distance” helps – they’re very closely linked, and, unlike for Euclidean distance and norm (which are very common concepts), few people will care about the one and not the

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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

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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.

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I reverted that change. It may be more streamlined but it is harder to read. I prefer the classical definition, and I added a brief sentence saying that first property is actually not necessary.

4515: 151: 4675: 2863: 5719: 1210:: "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." 3261: 4438: 5242: 4640:

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.

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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)

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It was my idea. I agree on the correctness of your formal equation. Maybe it is better to use it to avoid confusion about the power that has to be applied component-wise. --

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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.

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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

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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

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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.

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norms and normed vector spaces are essentially the same concept; in all likelihood, the reader who wants to learn about one wants to learn about the other as well.

5724: 5674: 5300: 5205: 5156: 5133: 3625: 1779:. I didn't confuse hamming distance with the taxicab norm. My interpretation is not incorrect, hamming distance and 1-norm are equivalent reductions(HAMMING 3683: 1245:

If you want to add more stuff to this article, please go ahead. But don't do much integration work by removing stuff from other articles. That seldom works.

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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.

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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.

1742:, with p < 1. this is not really a norm since the unit sphere is not convex. as p tends to 0, the unit circle probably looks more and more like a cross. 5734: 141: 233:

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

4248:. As you know, the two different symbols each have their own article, and each have what appear to be identical redirects that target those articles: 5704: 4147:{\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{p}={\frac {|x_{k}|^{p-1}\operatorname {sign} x_{k}}{\|\mathbf {x} \|_{p}^{p-1}}},} 3468:

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. —

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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

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That article is not very well suited for explaining the L2 norm. IMO it's worth having a separate article, as it's used quite commonly. Info from

4291:, a dab page that contains this article. Since the symbol is mentioned on this page, I thought of this page as the best target of the ones on the 5714: 4985: 1926:, 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 1718: 2440:

Isn't the "length" of the arrow determined by the norm? Is there anything really more fundamental about "length"? The statement seems circular.

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This formula is also valid for 0 < p < 1, but the resulting function does not define a norm, because it violates the triangle inequality.

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It's not obvious how to compute a complex norm, as you multiply by the complex conjugate, which means you essentially disregard the quantity

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We are on the same wavelenth. When I hear people taking big plans about reorganizing things I assume the worst. :) Happily not this time. :)

1099: 688: 5435: 4545: 4526: 2171: 1722: 488: 3607:). Indeed, for a field F, algebraic vector spaces over F are characterized by the cardinalities of their Hamel bases. For a vector space 1484:

article is different. There all the properties are consequences. Here are are talking about how to define an abstract concent like norm.

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The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude.

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Err, yes that is what I meant. I just thought to improve the expression as well as provide a relevant link for the last formula in the

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in a torus space (using the shortest path on the torus) doesn't fulfill the triangle equality, but the condition d(x,x)==0 <=: -->

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lateral clicks symbol. Thank you for finding the Unicode – I've added it and the HTML code to the paragraph. Hope this helps. –

3959:{\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{p}={\frac {x_{k}|x_{k}|^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.} 2189:

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.

5699: 5415: 253:. The properties of the norm function are given in the article. I do not know how to make this more explicit in the article. 1438:

Would it be okay to move this along with positive definiteness as another property that follows from the definitions? The

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Does anyone know what the modulus of an element of an arbitrary field is? Does the notion of modulus exist in every field?

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redirect to a section here that's prsently missing. Let's merge it in. This is an alternative to the proposal to merge

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all examples for norms given here are also examples of normed vector spaces, so we could have all examples in one spot.

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I would like to second this. The current article other wikipedia articles are not clear on pseudonorms and seminorms.

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I will try to answer your question "I don't know what the || notation means." by an example. If a=(3,4) then ||a||=

3680:, where A is an arbitrary (possibly infinite) set, we can consider any p-norm w. r. t. the basis, e. g. the 1-norm 2549: 2543: 2042: 655:

For a finite field, I think there is only one. Define |x|=1 if x is not zero, and |0|=0. Not terribly useful! --

2061:??? And where do you get this general concept of making astract concepts central? Seems like a bad idea to me. 3999:

There is the presentation question of whether the expression I used (only clearly usable for a zero component if

3140:(At least, this is true for finite-dimensional vector spaces; I'm not sure about the infinite-dimensional case.) 2079:

As Charles Matthews says, the uniform norm is a very important concept in analysis, and deserves its own article.

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here, since it is just a finitary special case, under the general principle of making abstract concepts central.

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symbols), not about what it is used to represent, but until we have that, I guess this is the best approach. —

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One of the key properties of norms is that they are all equivalent up to constant factors. In particular, if

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It's possibly a simple oversight, but complex numbers all have norms, or at least that was what Euler called

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A useful consequence of the norm axioms is the inequality ||u ± v|| ≥ | ||u|| − ||v|| | for all u and v ∈ K.

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There are a lot of names for various norms. We need some cleanup to clarify this. There's "Euclidean norm",

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I improved the Notation section and put it after Definition section which is a more logical place for it.

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x=x holds. Unfortunately these terms are totally mixed up in all the related articles here on wikipedia.

1800: 1780: 4417: 3176:, then invoke fact that continuous functions achieve their maxima and minima on a compact set to obtain C 1361: 1206:, norms based on the general Lebesgue integrals as well as countable sums, and sup norms? To quote from 443: 5213: 4944: 4905: 4866: 4206:, which I just worked with today and which prompted me to mention the confusion here in this article. – 3492: 2038: 1637: 1597: 1229: 1031: 50: 5086: 3800: 3544: 3143:(Proof in brief: prove that any norm is continuous, reduce to considering the unit ball by dividing by 1911: 94: 4443: 3146: 2993: 2960: 2230: 1633: 1593: 5681: 5208: 4978: