Talk:Inner product space

2910:, ...", I'd agree with you. Either change that (though I think you'd find that a hard sell: mathematicians might say it belongs in a mathematical discipline), or bring in notable references from general linear algebraists. The references are heavily skewed to Hilbert spaces and quantum mechanics; in fact, this article few references. Some mathematicians also tend to put normed spaces in primary position, so I understand that there will still be tension between the ideas even regarded within the scope of mathematics. The question becomes: within mathematics, how is the term used generally? I've provided a counterexample that suggests a presentation in which the concept of an inner product that naturally becomes positive-definite when applied in the context of a Hilbert spaces. In a way, this might be the usual tension between: do we present the familiar scope first, then give a "generalizations" section, or do we define the concept as the more general case, and then expand on the more familiar case? I think it comes down to what mathematicians generally consider the term to mean. — 1107:

course vector spaces (usually), so functions are vectors (and vectors are functions, in fact). I think that when you start using the vocabulary of "inner product space", you are usually in a case where almost everything is a vector and only very few objects are scalars, which you then denote by Greek letters since too much boldface or arrows would be more confusing than helpful. (Boldface is then sometimes rather used for matrices of scalars and the special case of coordinate "vectors", i.e. column matrices.) (In some rare cases, a little lack of notational consistency can be more pedagogical than too much notational rigidity.) However, this article is somewhere in the middle between these two worlds, and if you wish to put all vectors in boldface, please don't hesitate and do it! —

475:

either be defined in the mathematical way, or it should be made clear that this is a definition for physics and that mathematicians use another convention. As to the suggestion that this "convention seems to be working its way into recent mathematics", even if this is so, it is certainly not standard practice yet (for example, most university mathematics courses, and every textbook I have seen, would define the innner product as being linear in the 1st argument), and if I understand Knowledge's ethos correctly, the article should reflect current practice.

95: 85: 64: 31: 1578:@JBL: Thanks for your comments. Yeah, got it regarding "basis"; obviously I was intending the everyday meaning. Anyhow, it's still a mystery to me what "naturally" means. Does it just trivially mean "oh look, combined with square root we have Pythagoras, which is the length of a vector in ordinary geometry, and hence it's a useful norm"? Or "naturally" mean something more profound? 3530:, since such a title suggest that the content is an editorial synthesis. Indeed the second subsection of "Remark" (that is, "Base field") is pure original synthesis and does not belong to the article. On the other hand, the other subsection, called "Notation and the antilinear argument" deserve to appear in its own section, called "convention variant", with an improved writing. 22: 3613:

some stuff about additivity; I'm fine with this removal. But your new section on "alternative definitions" seems unnecessary and verbose. I could see having a sentence or two here about what a "hermitian form" is or whatever, but not three paragraphs. Actually the original version has the couple of sentences I'm envisioning, in the middle of "Elementary Properties".

1401:, and this link goes to an explanation - except that the section link is broken, and the linked article talks about the metric without saying that it's a metric, and in any case its talking about a metric induced by a norm, and no norm has been mentioned at the point of the cryptic link. So, yes, it's pretty bad, and a number of fixes are needed, both here and in the 1812:, which is the commonest and simplest instance of an inner product. That article is linked in the first paragraph, and I would expect anyone looking for it will find that article first – "dot product" is the name most often used when it's being studied at high school level. That does include numerical examples.-- 1600:"Naturally" is a term that seems to be used extensively in mathematics, meaning (I infer as a non-mathematician) that in there is some sense essentially only one way to do it. Sometimes, that means that there may technically be any number of ways to do it, but that all those ways are equivalent/indistinguishable/ 1835:

additional structure, such as a distinguished automorphism." No reason or reference is given for this assertion (either for the fact that additional structure is needed, or exactly what that additional structure has to be). "Distinguished automorphism" is not defined, and there is no link to a definition.

3533:

Here is an important issue: A non-expert reader must easily know which properties are specific to inner product spaces, and which are general properties of normed spaces. He must also clearly understand that giving an inner product is equivalent to give a definite Hermitian form. Presently, all these

1761:

I'm not really a mathematician, but this is very difficult to understand! I feel like this ought to be a simple operation, but it is very difficult to tell what the inner product is actually doing based on how the article is written. I'm sure everything is correct, but having never seen this before I

1525:

This article should be accessible to mathematicians and engineers alike, so perhaps it needs a bit of a rewrite. Perhaps the use of the term "space" could be avoided in the early stages of the decription (even though familiarity with the properties of vectors implies familiarity with a vector space,

482:

Frankly I dont see a point to the discussion in the first place. I dont know why this is a notable topic. Given the conjugate symmetry property and linearity in either term, you can easily prove linearity in the other term. Its arbitrary and moot. I get the sense that you guys are approaching this

3348:

I see a long Remarks section in 2. I think this content is interesting but it should not be at that place in that form. For the question about which position should be linear, the sentence or two in Note 1 seems sufficient, and if there should be more content about this, it should be further down,

2654:

As the inner product maps to the underlying field of scalars, and the complex field lacks an order relation, how does the inequality in the axiom on positive definiteness make sense? Should the left-hand side of the inequality be the _norm_ of the inner product of the vector with itself, instead of

1630:

Huh? So there's a thing called a pre-Hilbert space, which is incomplete, and has an inner product, but because it has an inner product it also has a norm, and because it has a norm it is "complete with respect to the norm", and therefore it's a Hilbert space. So according to this sentence, there is

1559:

It means that if you give me the inner product, I can use it to produce (in a natural way) a norm. Your two sentences together get close; the second sentence is problematic because of the use of "basis". This is explained more fully in the body of the article: "However, inner product spaces have a

1106:

I suppose nobody cares... if you work for some time in vector spaces (which often will be spaces of functions defined on other vector spaces), you sooner or later stop putting arrows on all these vectors - actually it would rather confuse me to see arrows on functions, but spaces of functions are of

666:

redirects to dot product, which seems inaccurate since the dot product is just an example of the inner product but is not synonymous. Also, the majority of the inner product space article describes the properties of the inner product. Wouldn't it make sense to move that content to a separate inner

3612:

I'm fine with merging the sentence "In this article, the field of scalars denoted F..." into the following sentence and making F mathBF. I generally don't like the other changes to "Definition" -- making it more complicated, adding the small/bold/right-justified/parenthetical labels. You took out

3548:

Ok. I agree the part about base field is OR, and the dot product is mentioned in the first paragraph of the lead already. I deleted them, and most of the stuff about bra-ket notation, leaving just a couple of sentences in a "convention variant" section, which mostly still needs to be written, but I

3391:

Just now, I remark that after his first revert, Mgkrupa did several edits that I reverted with my subsequent reverts. This includes the new section on completeness and Hilbert spaces; see below. Also, Mgkrupa has edited other parts of the article. I have not checked them, and thus I have no opinion

3333:

2 formats more of the math as html, e.g {{math...}}. I think the MOS policy is not to make sweeping changes to the math formatting without a good reason? So is D. Lazard's version the "original" here? Mgkrupa, do you want to argue for this change? Should we just agree to a policy for this article

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refers only to a specific case (but the informal verb "to dot with" can also be used with general inner products), so please do edit the articles to make this clear. I don't like short pages like the MathWorld inner product space page, unless they have potential for growth, and this seems to be the

3577:

contains a rough idea of the changes I'd like to see. I have not polished it yet (due to time) and any small mistakes would (of course) be fixed before adding it to the actual article. Parts of it also need to be reworded. One more thing that I'd like to include is an answer to the common question

2873:

Kaplansky goes on to describe a broadening of the theory to alternating and Hermitian forms, but it is not clear to me whether he is including these under the definition of inner product spaces, though he does indicate that it can make sense to include such generalizations from the start. So: is

2846:

As suggested by a few threads above, the restriction to positive-definiteness seems to be discipline-specific. Just because some fields (e.g. quantum mechanics) choose to restrict the types of inner products that they work with does not mean that their restrictions should be imposed on the whole

1777:

Perhaps you would like to look at euclidean space where the inner product really is a simple operation? And for not so simple inner products, consult Sobolev spaces where the inner product involves the weak derivatives of functions? In an abstract Hilbert space, the inner product really just is a

474:

I, and I belive most mathematicians, regard the inner product as being linear in the 1st argument. While I appreciate that this may be inconsistent with the way physicists use the inner product, the article seems to be written as if it were from a mathematical viewpoint, and so I think it should

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In addition to this the definition of non-degeneracy is misleading: a non-degenerate inner product is one for which the matrix that generates it is non-singular. An hermitian matrix of signature (n,1) has an entire subspace of zero vectors in C^{n+1}, but the hermitian form is non-degenerate. The

1249:

Although it is hinted in the article that inner product (over Euclidean spaces) is related to intuitive definition of angle, and infact the inner product is used to define the angle (in the subsection "Norm"), it would be beneficial to show that this indeed corresponds in Euclidean spaces to the

453:

Though I am used to Sequilinearity as measing linear in 1st argument, most physicists use the other convention. I have edited some articles with this assumption; moreover, many of the quantum mechanics articles naturally follow the physicist's convention. At this point the physicist's convention

3616:

It doesn't make any sense for the first mention of "Hilbert" on this page to be a non-Hilbert example without explaining what a Hilbert space is. I think the 1-d examples should be cut and the examples should be: R^n, C^n, an actual Hilbert example (starting with one sentence explaining that a

1665:

Thanks for the reply, and sorry about the delay on my part. The key is that the norm is used in the act ("completion") of creating a new space whose properties qualify it to be a Hilbert space. This is obfuscated by the current wording "becomes a Hilbert space". There is actually no space that

1543:

In the intro: "An inner product naturally induces an associated norm" What do the words "naturally" and "induces" mean in this context? Is it possible to "unnaturally induce" something? Does this sentence translate in plain english simply to: "An inner product implies an associated norm", or

1453:

This article is confusing without some prior knowledge of what the inner product is by itself. I can see how the concept of an inner product space is closely related to that of an inner product... but this article seems to be written with the assumption that the reader already knows about inner

3507:

I see. I have changed the Rs and Cs. I'm confused about the Remarks section. You complained about it but Mgkrupa said he did not add it (he just reorganized it into a higher-level section), and your version (10:28, 22 December) has essentially the same content in an "Alternative definitions,

3355:

2 has a properties section at the top of "Basic results, terminology, and definitions" about L-semi-inner-products and Hamel basis, which are not topics I'm familiar with. This definitely shouldn't be the first subsection of this section, but I don't have an objection to it being moved to the

1502:

If the vast majority of visitors to this article are already familiar with vector spaces, then this is probably not a problem. (Since getting at the definition of the inner product becomes a relatively simple process of elimination in this case.) Maybe I am just a bit too far from my field.

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at least under the examples and basic properties. The part about the base field can be discussed (ideally more concisely) in the section about "Real and complex parts of inner products", or perhaps immediately before that subsection. But it shouldn't delay the examples and basic properties.

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In the "Remark", under the "Definition", it is stated that it is necessary to restrict the basefield to R or C. This statement is contradicted in the same paragraph by the statement that any quadratically closed subfield of R or C will suffice. It is stated that "The basefield has to have

2312:, expressing the preference to write the coefficient before the vector and to have the two occurrences of the basis vector close together. For this to work one needs complex linearity in the first argument. However, from a matrix point of view, this can be most consistently rewritten as 605:

Could we add an explanation on how the inner-product is related to Dirac notation? Perhaps I am requesting a discussion on the notion of dual spaces and inner-products. In "Principles of Quantum Mechanics," Shankar mentions that one can think of the inner-product as a mapping from

944:), as it's often important for a system to be able to perform this particular operation quickly -- system performance is sometimes measured in "dot products per second". I'd be in favour of rewording the intro to say that these things are related, and not simply the same thing. 3654:, for one). The elaboration of positive definite into definite and positive semi-definite is unnecessary. The discussion of Hermitian forms is still too long. Maybe a sentence could be added about the alternative definition of an inner product space, but not two paragraphs. 1476:

It seems to me that the article can hardly be improved in this respect. Defining an inner product cannot be done without first defining vectors and a vector space, and only then defining the the inner product, upon which the article focusses in its lead and first section

1121:

They really should put the vectors in boldface to distinguish them from scalars. I know that mathematicians often don't do this, but there is no good reason not to. Otherwise beginners may be confused about which things are vectors and which things are scalars.

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naturally defined norm based upon the inner product ...." Keeping in mind that this is just an introduction and the details are explained in the body, do you think there's a good way to clarify this jargon that doesn't make the sentence too much of a mess? --

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issue from the perspective of a physicist who only knows implementation, rather than the perspective of a mathematician who abstracts. Im less concerned with what is more popular and would prefer to default definition to the historical inspiration.

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From the definitions of inner product and bilinear operator, follows that the inner product over a complex vector space isn't a bilinear form (but it is for real vector spaces). So, the definition in the article contains a wrong generalization:

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on which edits are improvements, and which are not. This is because I do not want to spent too much time on this article, and if the lead and the definition section are correct, the remainder of the article is less important for most users.

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Doesnt the field the inner product maps to have to be an ordered field? What kind of order? Doesnt make sense to talk about inequalities (positive definiteness) in a non-ordered field, so I dont know why complex numbers are an appropriate

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think it's better as it is than with all the confusing or unnecessary content that was there before. (If you think I overdid it, I'm happy to add back the stuff about bra-ket notation; it just seemed too confusing to be useful as it was.)

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I know that a lot of mathematicians do not use an arrow to denote a non-spacial vector, but in just about every textbook I've read, vectors (even non-spatial ones) are always boldfaced. Is this something we should change, or does nobody

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products. There is admittedly some mention of what an inner product does, but I have found no real explanation of what it is. It seems like there should be a separate article for "Inner Product"... or at least a section explaining it.

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blackbox where you put two vectors (points in a vector space, not necessarily column vectors) in and get a real or complex number out. With certain restrictions on linearity etc. that allows, e.g., to define the angle of two vectors.--

1005:

I guess I'm the one responsible for that glaring ommission. All I can say in my defense is that I had a complete rewrite of this article in a browser window which crashed, completely draining my impetus. I guess I have to fix it.

3366:

My suggestion is to use 1 as a baseline and try to reach a compromise from there. Mgkrupa, are there other edits or specific disputes that I'm missing that way? Is there another version in the history that I should be looking at?

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without first mixing it with that of a vector space. It was admittedly pretty easy to extract the definition of an inner product from this article once I had read the Wolfram equivalent, but only because I knew what to look for.

3388:), and revert Mgkrupa's reverts. Otherwise, I have not edited the article recently. This does not mean that I find the section and the remainder of the article perfect. Only that the old version is much better than Mgkrupa's one. 1738:

I dont know if the article has been changed since you made this comment, but it currently reads that F denotes the field of reals or complex, specifically in this article... not that the field always has to be reals or complex.

3310:

It is very difficult to provide a third opinion on this dispute because there are literally hundreds of edits involved. I will try to provide an opinion on the differences I see between the last two versions of the article

1719:

The definition given in MathWorld isn't even correct for a vector space over the complex numbers (since axiom 3 is wrong), and for an abstract base field it's meaningless (since the expression used in axiom 4 is undefined).

1493:

Thanks for the answer, though I still partially disagree. A basic understanding of vectors is certainly necessary. I'm not so sure about vector spaces. MathWorld does a pretty good job of giving the basic concept behind an

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Agreed. There's some confusion, but I think the physicist's convention is the one to follow, since it has a good reason for using that order. The convention seems to be working its way into recent mathematics, as well.

3639:) suggestions. I'm fine with cutting the 1-d examples and your suggestions about how to introduce Hilbert spaces. But there should be a discussion somewhere about how an inner product space relates to its completion. 2413: 2193: 1142:

I suggest changing "An orthonormal basis for an inner product space V is an orthonormal sequence whose algebraic span is V." to "An orthonormal basis for an inner product space V is an orthonormal sequence

3478:, and Mgkrupa added recently a section "Completeness and Hilbert spaces". IMO, both are badly written, and they deserve to be merged. However, this may hardly be done without cleaning up the awful section 2738:

Why must the inner product map to reals or complex numbers at all? Why cannot F be a general field? Do we have a generalized definition for inner products or does the field have to be specific to reals?

1646:

You are correct until "and because it has a norm". Rather, because it has a norm it can be completed in this norm to give a different space called its completion, and the completion is a Hilbert space.

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2 has a sentence about dot products at the start of "some examples". This seems like it should be moved into the Euclidean space subsection, where there should be an example of using the dot notation.

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that predated his edits, because it was correct and clearer, while Mgkrupa version was confusing by hidding the main facts behind comments that are either out of scope, or original research, or both.

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From the intro: " An incomplete space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm, induced by the inner product, becomes a Hilbert space."

1063: 3165:"motivations" is gone. The information in the "remarks" section was not added by me; it was there before I edited this article. (added later: to be clear, I'm referring to my latest version 151: 3617:

Hilbert space can be infinite-dimensional but it must be complete), and then maybe the non-Hilbert example. I think then the "completeness and hilbert spaces" section will be unnecessary.

3128:, it is to you to show that your edits improve the article. In particular, it is to you to provide sources that show that the discussions referred to as "remarks" or "motivations" are not 2874:

anyone going to claim that Kaplansky is not a notable secondary source? By implication, unless the contrary is demonstrated, the article must be updated to the more general definition. —

879:, and the definition of dot product given in the Examples section. Also, I know computer scientists sometimes talk about dot product, referring to the "specific example" definition (i.e, 2933:

Agreed on precedence. In this context, I would like to see this as "most common amongst mathematicians", or at least generally. At the moment, references seem to be skewed to physics. —

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would already have to be a Hilbert space. And in a Hilbert space, the surjectivity of the map is actually the content of the Riesz representation theorem, not part of the definition.

3089:"editing the article again must be avoided without explaining the reasons of the edit (never done by Mgkrupa), and without getting a consensus on the talk page." Same applies to you. 3244:

I agree to not edit this article, except to revert edits that are not a clear improvement or are not be validated by a consensus in this page. If you revert my revert, this will be

767:. At least that's what I learned. Maybe splitting the article isn't the best solution, but I feel that some clarification is needed at least in the beginning paragraphs. Perhaps 2700: 2959:

Dropped the reference to the semi-inner product. Eliminating the 2nd requirement in the Positive-definite definition does not agree with the wiki definition of semi-inner product.

2517:

The remark is at the end of a list of lemmas that follow from the 3 axioms. I - and I believe many readers will do the same - assumed the actual definition ended with the line:

1198:

The Cauchy-Schwarz proof may be important, but on the "mathematician's" conventions adopted on this page, it seems to me to be wrong. \lambda should be defined as <y,y: -->

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Also, it seems that we should mention another common notation for inner-products: (x,y). In fact, the page on bra-ket notation refers to the inner-product in this manner. (

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May I suggest moving the caveat in among the list of axioms or immediately after, and splitting off the discussion of lemmas into a subsection captioned "Basic lemmas"?

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This section mentions several times that it's possible to induce a metric from an inner product, but neither this page nor the page on metrics shows this relation...

3009:. He also restored or added inacceptable edits such as the introduction of grammatical errors (removing the article before "inner product space"), unusual notation ( 2591: 1688:

The definition in this article starts with: "In this article, the field of scalars denoted F is either the field of real numbers or the field of complex numbers."

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for a partially unspecified field" Is there a policy against this notation? I am not aware of any. Is this is a problem then this can be fixed by find+replace all.

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even if not necessarily under that name); maybe merely referring to vectors would be adequate. Anyone want to tackle wording in this article to address this? —

1854:

According to the definition a property of an inner product is "Linearity in the first argument". In the example section there is an example of an inner product:

677:

Let me start by saying that it is nice to see some fresh blood. Your contributions to the linear algebra articles are much appreciated. Are you already aware of

3522:

My opinion is that, before Mgkrupa's edits the article had many serious issues, that are not fixed by Mgkrupa's edits. Calling a section "Remark" goes against

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Of course, if one wants to weight the points of view, the mathematical one is slightly more "wrong". Mathematicians like to write the basis decomposition as

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Why do you want to split this article? I think it would be rather difficult to do it cleanly, and the article is not that big that it needs to be split. --

3806: 3053:"introduction of grammatical errors" Oops. A minor problem with an easy fix. Everyone makes a minor mistake here and there. (e.g. you misspelled "hidding") 594:

I have never seen a definition of inner product which isn't linear in the first argument. I am a mathematician and consider the current version incorrect.

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GTM 135 uses exactly the definition given in the article. Edit: As does GTM 96. I don't think "most general" has automatic precedence over "most common".

1793:

I'm having trouble with it, too. If it's possible to give an example of an inner product at its simplest, using numbers, I suspect it would help a lot.

2415:, where clearly the coefficient comes after the basis vector, and to have the basis vectors typographically close together the scalar product should be 837:

Here is my site with inner product example problems. Someone please put this link in the external links section if you think it's helpful and relevant.

3776: 3578:"What are complex inner products conjugate symmetric?" I'll do these changes and also respond to some comments/questions above once I have more time. 3056:"ordering the axioms of inner product spaces by numbering them" Is there a policy against this? I am not aware of any. If so them change <ol: --> 3786: 2236: 2012: 1214: 3657:

The definition of a Hilbert space shouldn't come before the Euclidean space examples. Defining a Hilbert space as a Banach space is unhelpful.

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I still think most of the changes you're proprosing to the definition section make it worse. The overuse of bold and italics is not good (see

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properties are mixed in a confusing and unstructured way. This was already the case with the older version, but is worse after Mgkrupa edits.

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The article uses the pure math textbook convention expressed in notation commonly used only among physicists, this may confuse some readers.

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Did You read the remarks at the end of the definition, starting with "Some authors, especially in physics and matrix algebra, prefer..."? --

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Could we have a paragraph or so regarding the history of the inner product? When and by whom it was defined, and in what context? Thanks.

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I think that this example is incompatible with the definition. I believe that in order to be linear in the first argument, it should read

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I thought those things were synonymous. I did not hear of the notion of "inner product" in other settings. 23:06, 19 February 2006 (UTC)

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This definition is too narrow. Inner products may be negative or zero. The definition given is for a positive definite inner product.

2740: 1740: 1510: 484: 194: 178: 3317:, henceforth 1 and 2; 1 is the current version and D. Lazard's version). Here are some differences, including those discussed above: 3162:) for lists, and to many changes that are individually minor, but all together make the article harder to read for non-specialists. 738: 678: 434: 108: 69: 3086:

Some of the proofs I took from the references that I added. I add those proofs for readers who might be interested in reading them.

3771: 2063: 1294: 507:"Formally, an inner product space is a vector space V over the field F together with a bilinear form, called an inner product" 3136:

states that if a style is acceptable, it is forbidden to change it without explicit strong reason. This applies to the use of

3113:. If you only list trivial objections OR if you do not response in a reasonable amount of time then I will revert your revert. 875:

I was under the impression that the dot product was a specific example of an inner product -- that seems most consistent with

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which is handled with the bilinear form concept even if there is a tradition of persisting in the inner product terminology.

1604:. So yes, it does mean something more profound, but I have yet to find a mathematically precise definition of the term. — 1320:: some symbols appear as empty boxes. I went into edit to see which HTML character it was; but it also appeared as a box. 3129: 44: 3359:

2 has a section "Completeness and Hilbert Spaces". This seems fine to me. D. Lazard, do you have something against it?

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has an extensive list of HTML characters including ⟨ (&lang;). COuld you fix it this way? I don't have the time.

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treated as synonomous. In Euclidean space they often are treated as synonymous but in the general definition, the

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have no idea what it is talking about. Could someone make the language a little clearer and easier to understand?

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You're right. The article at one time used the physicist convention and during that time the proof was included.--

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the reader should probably be cautioned that this is different from the convention common to physics textbooks:

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This would depend on the majority view of the discipline in which it is presented. If the article started "In

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The symbols following the word "map" are unfamiliar to me. Can a link to an explanation of them be supplied?

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must map to an ordered subfield of the complex then, namely (most likely) the reals? This was not specified.

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no difference between a Hilbert space and a pre-Hilbert space, except perhaps the state of the explanation?

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There is no restriction to positive-definiteness, not even to nondegeneracy. A degenerate example is given.

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Which symbols? The angle brackets? Or the mapsto arrow? Or the colon? The angle brackets are a conventional

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notations and remarks" section (plus the sentence about dot products). Did you want this content removed?

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formerly was not a Hilbert space, and then presto "becomes" a Hilbert space. Thanks for clearing that up.

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I think the article is in a bad state, and it needs to be restructured, but for now, I've given the map. -

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for a partially unspecified field), ordering the axioms of inner product spaces by numbering them, etc.

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if the product is linear in the first argument, and it is an antilinear isomorphism, not an isomorphism.

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bottom. (D. Lazard, do you want to specifically argue against this content being anywhere on the page?)

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Anyway, I think requiring it to be an iso is too strong because the dual space is always complete, so

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should not be a redirect at all but have some content that explains the axioms and properties? Maybe

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Fair enough. But what is then the thing called? John B. Conway calls it a semi-inner product in his

1990:{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle :=\mathbf {y} ^{*}\mathbf {M} \mathbf {x} .} 1916:{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle :=\mathbf {x} ^{*}\mathbf {M} \mathbf {y} .} 1544:

perhaps "An inner product provides a basis for calculating a norm" or something along those lines?

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notation. Everything else comes from the conventional symbols that define mathematical functions. —

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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An arbitrary base field is permitted. Finite fields, including of characteristic 2, are covered.

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Mgkrupa has essentially restored his version and moved his editorial comments to a new section

857:. Are all inner products dot products, or is the dot product an example of an inner product? -- 3651: 2973: 2934: 2924: 2911: 2897: 2875: 2816: 2707: 1725: 1610: 1530: 1485: 1410: 818: 722: 708: 630: 568: 512: 464: 3330:

1's version of the definition section is better: unordered lists, no bold labels on the right

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Please list your other objections/suggestions. And please refrain from any more edit warring.

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to the change of list format from bulleted to numbered, to the use of html tags (<ol: -->

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describes for general inner products. The articles can be much improved, though, especially

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Moreover, it doesn't say that the map is in some way associated to the sesquilinear form.--

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Update: Im assuming that positive definiteness between an element and itself <x,x: -->

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Wait what? In any Hilbert space, the basis has to be at least continuum in cardinality.

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is the dual space of bras (obtained by taking the conjugate transpose of the kets).

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I'm not criticising you lethe, I'd probably be annoyed too after a browser crash :)

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You have not replied to any of my responses. List what objections you have against

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To return to your question: it seems from the top of both articles that the terms

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says: "This definition also applies to an abstract vector space over any field."

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I have a similar problem. I guess the symbols ought to be brackets or something.

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You can surely add one more section to the article explaining the connection to

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is the term to use instead of "indefinite inner product".My primary concern was

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Remarks about using fields other than C or R are inadequate and contradictory

2028: 787:? Personally I like the way MathWorld has these topics organized with an 1108: 3381:

For the record my recent edits consisted only to restore the version of

2408:{\displaystyle v=(e_{1},\dots ,e_{n})\cdot (x_{1},\dots ,x_{n})^{\top }} 2188:{\displaystyle \langle ax,y\rangle ={\overline {a}}\langle x,y\rangle .} 3133: 2907: 692:

are treated as synonymous. I actually quite like the current division:

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http://www.exampleproblems.com/index.php/Linear_Algebra#Inner_Products

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did a lot of edit on this page. Yesterday, I restored the version of

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section? What he calls an inner product, most everybody else calls a

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Can this be strengthened? For Hilbert spaces the following is true:

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as at present. Then, for example, a term like <\lambda y, x: -->

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Good point. We could split it into two cases, or we could rewrite

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an inner product (also called scalar product or dot product), ...

2488:, which would need complex linearity in the second argument.-- 2047:

When the current version of the article states the following:

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usual definition of angle given by arccos of base/hypotenuse.

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an inner product (also called scalar product or dot product),

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describes the standard inner product on Euclidean space and

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defines inner product spaces clearly in the general sense:

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Positive definiteness and the order relation on the scalars

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Fixed, please see my edit. There is no need to write about

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is indeed taken to mean a positive-definite bilinear form.

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That is a very context-dependent point. In many contexts,

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That sentence is necessary to introduce the "variable" F.

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is not real-valued. I changed the article accordingly. --

3470:

About Hilbert spaces: There is a longstanding subsection

2481:{\displaystyle x_{k}=\langle e_{k},v\rangle =e_{k}^{*}v} 1158:

which has V as the smallest closed subspace containing {

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According to preliminaries also: 0 =< (<x, y: -->

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http://planetmath.org/encyclopedia/NonDegenerate.html

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Finally, per axioms 0 =< <(x + y), (x + y): -->

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Confusion between conventions in math and in physics

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I will change the article, correct me if I'm wrong.

112:, a collaborative effort to improve the coverage of 3685:Anyone opposed to removing the following sentence? 3248:, and this may lead you to be blocked for editing. 3449: 3424: 3327:I think R and C should be mathBB, F should not be. 3218: 3198: 3153: 3075: 3023: 2866:An arbitrary symmetric bilinear form is permitted. 2694: 2585: 2559: 2480: 2407: 2304: 2187: 2111: 1989: 1915: 1449:Why does "Inner Product" redirect to this article? 1057: 936: 737:Wow, speedy resposes. Thanks for directing me to 1058:{\displaystyle x\mapsto \langle \cdot ,x\rangle } 654:It seems to me that the current structure of the 2759:Strengthening to separability ⇔ countable basis? 3792:Knowledge level-5 vital articles in Mathematics 3001:, that is inacceptable in Knowledge because of 1173:. Equivalently, its closed linear span is V."-- 3226:"? Are we in agreement to replace "<ol: --> 2886:Alternative definitions, notations and remarks 2670:Note that the conjugate symmetry implies that 1700:http://mathworld.wolfram.com/InnerProduct.html 3480:§ Basic results, terminology, and definitions 1691:But isn't an inner product space defined for 8: 2689: 2677: 2548: 2536: 2454: 2435: 2286: 2267: 2179: 2167: 2151: 2136: 2103: 2091: 2082: 2067: 1956: 1940: 1882: 1866: 1683: 1052: 1040: 1808:I think the article you are looking for is 406:i.e. || x + y || =< || x || + || y ||. 966:The canonical map isn't defined anywhere. 58: 3440: 3439: 3437: 3418: 3417: 3415: 3211: 3189: 3188: 3186: 3144: 3143: 3141: 3069: 3068: 3066: 3017: 3016: 3014: 2675: 2572: 2560:{\displaystyle \langle x,x\rangle \geq 0} 2534: 2469: 2464: 2442: 2426: 2420: 2399: 2389: 2370: 2351: 2332: 2317: 2296: 2280: 2261: 2250: 2238: 2157: 2134: 2065: 1979: 1974: 1968: 1963: 1951: 1943: 1938: 1905: 1900: 1894: 1889: 1877: 1869: 1864: 1203:can be rearranged as \lambda <y,x: --> 1032: 923: 917: 911: 901: 890: 884: 563:. At least the standard inner product on 259:Priliminaries, directly from the axioms: 3034:So, I have reverted all this stuff. Per 1027:First a minor detail, the map should be 763:synonymous and is just an example of an 539:I think the inner product map should be 3782:Knowledge vital articles in Mathematics 60: 19: 817:general opinion here on Knowledge. -- 368:)^2. Together with axioms, therefore 3797:B-Class vital articles in Mathematics 2514:), thanks for your informative reply. 1209:|^2, as required later in the proof. 454:seems the one to follow in Knowledge. 337:Cauchy-Bunyakovski-Schwarz inequality 254:Cauchy-Bunyakovski-Schwarz inequality 7: 3266:I would also like to point out that 2954:An edit summary from a recent edit: 252:is obtained as a consequence of the 106:This article is within the scope of 3181:So are we in agreement to replace " 2884:Why not just list Kaplansky in the 2695:{\displaystyle \langle x,x\rangle } 2247: 986:vector space, it suffices to check 49:It is of interest to the following 3807:High-priority mathematics articles 2400: 1397:is followed by a link on the word 812:I think it is correct to say that 14: 1393:The first occurrence of the word 739:Knowledge:WikiProject Mathematics 679:Knowledge:WikiProject Mathematics 248:To illustrate that, and how, the 126:Knowledge:WikiProject Mathematics 3777:Knowledge level-5 vital articles 3593:. Any suggestions or objections 1980: 1975: 1964: 1952: 1944: 1906: 1901: 1890: 1878: 1870: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 1245:Relating to definition of angle 853:In the first paragraph we have 601:Connnection to Bra-Ket Notation 400:Sqrt( <(x + y), (x + y): --> 146:This article has been rated as 3787:B-Class level-5 vital articles 3589:My proposed changes are here: 2966:A course in fuctional analysis 2396: 2363: 2357: 2325: 1037: 799:page. Just some thoughts... - 241:23:26, 13 September 2006 (UTC) 1: 3755:16:03, 26 December 2021 (UTC) 3731:09:40, 22 December 2021 (UTC) 3667:05:39, 30 December 2021 (UTC) 3646:02:20, 30 December 2021 (UTC) 3627:00:29, 30 December 2021 (UTC) 3608:23:37, 29 December 2021 (UTC) 3585:01:05, 28 December 2021 (UTC) 3559:23:30, 26 December 2021 (UTC) 3544:21:35, 26 December 2021 (UTC) 3518:16:02, 26 December 2021 (UTC) 3496:10:03, 24 December 2021 (UTC) 3450:{\displaystyle \mathbb {C} .} 3377:14:16, 23 December 2021 (UTC) 3306:14:37, 22 December 2021 (UTC) 3258:14:37, 22 December 2021 (UTC) 3238:14:45, 22 December 2021 (UTC) 3199:{\displaystyle \mathbb {F} ,} 3176:15:35, 22 December 2021 (UTC) 3154:{\displaystyle \mathbb {F} ,} 3120:13:52, 22 December 2021 (UTC) 3096:09:33, 22 December 2021 (UTC) 3048:09:22, 22 December 2021 (UTC) 2726:21:35, 25 November 2013 (UTC) 2712:13:27, 25 November 2013 (UTC) 2665:05:31, 25 November 2013 (UTC) 2645:04:57, 25 November 2013 (UTC) 1788:18:13, 26 February 2013 (UTC) 1772:18:08, 26 February 2013 (UTC) 1730:13:03, 18 November 2012 (UTC) 1714:11:28, 18 November 2012 (UTC) 1613:14:08, 1 September 2012 (UTC) 1588:11:27, 1 September 2012 (UTC) 1441:00:54, 22 February 2011 (UTC) 1415:16:32, 21 February 2011 (UTC) 1387:13:12, 21 February 2011 (UTC) 1361:21:56, 20 December 2009 (UTC) 1342:21:35, 20 December 2009 (UTC) 1116:23:03, 4 September 2007 (UTC) 867:16:45, 19 February 2006 (UTC) 662:articles could be improved. 443:14:54, 17 December 2007 (UTC) 373:) =< 2 Sqrt( <x, x: --> 120:and see a list of open tasks. 3802:B-Class mathematics articles 3425:{\displaystyle \mathbb {R} } 3167:here 09:38, 22 December 2021 3076:{\displaystyle \mathbb {F} } 3024:{\displaystyle \mathbb {F} } 2847:subject of linear algebra. 2837:17:13, 27 October 2020 (UTC) 2623:15:37, 30 October 2013 (UTC) 2498:10:18, 30 October 2013 (UTC) 2227:10:04, 30 October 2013 (UTC) 2212:09:37, 30 October 2013 (UTC) 2162: 1749:20:21, 6 November 2018 (UTC) 1684:Shouldn't it be "any field"? 1431:does the same trick better. 1138:Orthonormal basis definition 929: 650:Separate inner product page? 493:17:52, 6 November 2018 (UTC) 203:20:17, 6 November 2018 (UTC) 187:20:12, 6 November 2018 (UTC) 3111:this version of the article 2978:11:53, 8 January 2016 (UTC) 2851:Linear Algebra and Geometry 2027:I'm affraid you are right! 1676:00:37, 28 August 2012 (UTC) 1657:01:30, 23 August 2012 (UTC) 1641:21:26, 18 August 2012 (UTC) 1623:pre-Hilbert space confusion 1570:01:29, 23 August 2012 (UTC) 1554:21:18, 18 August 2012 (UTC) 1188:12:56, 9 January 2007 (UTC) 1183:I have put it another way. 1178:11:10, 9 January 2007 (UTC) 982:* is an isomorphism. For a 744:I guess my problem is that 614:rather than a mapping from 401:) =< Sqrt( <x, x: --> 382:=< 2 Sqrt( <x, x: --> 3823: 3681:Removed redundant sentence 2983:Large-scale edits reverted 2749:19:18, 21 April 2019 (UTC) 1303:12:19, 3 August 2009 (UTC) 1239:18:21, 18 March 2007 (UTC) 1227:16:16, 18 March 2007 (UTC) 1194:Cauchy-Schwarz Proof Error 1020:05:49, 24 April 2006 (UTC) 1011:04:24, 24 April 2006 (UTC) 833:Vote for new external link 741:, I wasn't aware of it. 390:=< (Sqrt( <x, x: --> 360:)^2 =< 4 <x, x: --> 3276:the the three-revert rule 2938:21:44, 27 July 2014 (UTC) 2929:19:04, 27 July 2014 (UTC) 2915:18:49, 27 July 2014 (UTC) 2902:18:24, 27 July 2014 (UTC) 2879:16:56, 27 July 2014 (UTC) 2849:Irving Kaplansky (1969). 2821:13:40, 3 April 2014 (UTC) 2780:has an orthonormal basis. 2655:simply the inner product? 1824:19:43, 2 March 2013 (UTC) 1803:19:05, 2 March 2013 (UTC) 1274:20:22, 31 July 2009 (UTC) 1255:13:11, 13 July 2007 (UTC) 1100:01:05, 24 July 2006 (UTC) 1085:19:17, 21 July 2007 (UTC) 1000:18:05, 9 March 2006 (UTC) 954:11:00, 18 June 2008 (UTC) 827:18:35, 22 July 2005 (UTC) 804:22:30, 20 July 2005 (UTC) 730:21:17, 20 July 2005 (UTC) 717:20:41, 20 July 2005 (UTC) 672:20:10, 20 July 2005 (UTC) 467:18:28, Oct 23, 2004 (UTC) 458:14:07, 11 Aug 2004 (UTC) 420:02:13 Mar 3, 2003 (UTC). 389:<(x + y), (x + y): --> 145: 78: 57: 3410:should be replaced with 3278:with these three reverts 2037:20:25, 12 May 2013 (UTC) 2017:17:49, 12 May 2013 (UTC) 1200:rather than <y,y: --> 801:Vanished user 1029384756 669:Vanished user 1029384756 586:03:37, 14 Jan 2005 (UTC) 571:12:00, 13 Jan 2005 (UTC) 532:01:00, 13 Jan 2005 (UTC) 523:07:25, 12 Jan 2005 (UTC) 303:) <1, (<x, y: --> 225:02:17, 8 Oct 2004 (UTC) 152:project's priority scale 3703:is either the field of 3631:I made some changes to 3294:10:28, 22 December 2021 3289:08:50, 22 December 2021 3284:21:18, 21 December 2021 2890:symmetric bilinear form 2716:Ha ha, okay, thank you. 2567:with equality only for 1845:16:03, 2 May 2013 (UTC) 1533:13:02, 2 May 2012 (UTC) 1519:07:51, 2 May 2012 (UTC) 1488:11:24, 1 May 2012 (UTC) 1470:09:14, 1 May 2012 (UTC) 1132:01:14, 2 May 2013 (UTC) 645:15:30, 5 Jun 2005 (UTC) 582:Oops, you're right. -- 402:) + Sqrt( <y, y: --> 391:) + Sqrt( <y, y: --> 293:) <1, (<x, y: --> 109:WikiProject Mathematics 3772:B-Class vital articles 3573:Hi everyone. The page 3451: 3426: 3220: 3200: 3155: 3077: 3025: 2696: 2587: 2561: 2482: 2409: 2306: 2189: 2113: 2054:in the first argument: 1991: 1917: 1367:Connection with metric 1206:\overline{<x,y: --> 1059: 938: 906: 424:correct definition is 3689:In this article, the 3452: 3427: 3221: 3201: 3169:; not to your version 3156: 3130:WP:Original synthesis 3078: 3026: 2697: 2588: 2562: 2483: 2410: 2307: 2190: 2114: 1992: 1918: 1217:comment was added by 1060: 939: 886: 353:=< 4 <x, x: --> 351:)^2 + 4 <x, y: --> 344:=< 4 <x, x: --> 36:level-5 vital article 3633:User:Mgkrupa/sandbox 3591:User:Mgkrupa/sandbox 3575:User:Mgkrupa/sandbox 3436: 3414: 3334:and make it uniform? 3210: 3185: 3140: 3065: 3013: 2857:Inner product spaces 2774:inner product space 2674: 2586:{\displaystyle x=0.} 2571: 2533: 2419: 2316: 2237: 2133: 2064: 1937: 1863: 1850:Incompatible example 1031: 883: 795:page, and a minimal 132:mathematics articles 3459:There is a section 3227:" with "<ul: --> 2811:orthonormal basis. 2474: 2266: 1539:"Naturally induces" 1403:normed vector space 1316:does not render in 1090:Vector Notation (?) 797:Inner Product Space 781:inner product space 773:inner product space 771:should redirect to 698:inner product space 656:inner product space 250:Triangle inequality 3447: 3422: 3216: 3196: 3151: 3073: 3021: 2950:Semi-inner product 2692: 2583: 2557: 2478: 2460: 2405: 2302: 2246: 2185: 2109: 1987: 1913: 1208:^{-1}|<x,y: --> 1055: 984:finite-dimensional 934: 775:instead? Perhaps 384:) + <x, x: --> 281:<(<x, y: --> 270:<(<x, y: --> 101:Mathematics portal 45:content assessment 3395:This being said, 3219:{\displaystyle F} 3058:. Problem solved. 2165: 2020: 2005:Panagiotis.niavis 2003:comment added by 1817: 1757:Difficult to Read 1509:comment added by 1460:comment added by 1377:comment added by 1339: 1230: 1205:^{-1}<x,y: --> 1201:^{-1}<y,x: --> 1199:^{-1}<x,y: --> 932: 791:page, a separate 667:product article? 515:to also apply to 513:sesquilinear form 445: 433:comment added by 380:) + <x, x: --> 323:)^2 <1, 1: --> 283:), (<x, y: --> 272:), (<x, y: --> 166: 165: 162: 161: 158: 157: 3814: 3744: 3728: 3720: 3712:or the field of 3711: 3702: 3643: 3605: 3582: 3506: 3456: 3454: 3453: 3448: 3443: 3431: 3429: 3428: 3423: 3421: 3409: 3403: 3303: 3273: 3235: 3225: 3223: 3222: 3217: 3205: 3203: 3202: 3197: 3192: 3173: 3160: 3158: 3157: 3152: 3147: 3117: 3108: 3093: 3082: 3080: 3079: 3074: 3072: 3030: 3028: 3027: 3022: 3020: 2854: 2806: 2796: 2791:A Hilbert space 2779: 2701: 2699: 2698: 2693: 2615:Cool dude ragnar 2592: 2590: 2589: 2584: 2566: 2564: 2563: 2558: 2487: 2485: 2484: 2479: 2473: 2468: 2447: 2446: 2431: 2430: 2414: 2412: 2411: 2406: 2404: 2403: 2394: 2393: 2375: 2374: 2356: 2355: 2337: 2336: 2311: 2309: 2308: 2303: 2301: 2300: 2285: 2284: 2265: 2260: 2204:Cool dude ragnar 2194: 2192: 2191: 2186: 2166: 2158: 2118: 2116: 2115: 2110: 2019: 1997: 1996: 1994: 1993: 1988: 1983: 1978: 1973: 1972: 1967: 1955: 1947: 1922: 1920: 1919: 1914: 1909: 1904: 1899: 1898: 1893: 1881: 1873: 1813: 1706:Klaas van Aarsen 1521: 1472: 1389: 1349:User:KSmrq/Chars 1340: 1338: 1336: 1328: 1322: 1212: 1207:} = <y,y: --> 1185:Charles Matthews 1064: 1062: 1061: 1056: 943: 941: 940: 935: 933: 928: 927: 918: 916: 915: 905: 900: 639:bra-ket notation 633:forgot to sign) 428: 209:Negative or zero 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 3822: 3821: 3817: 3816: 3815: 3813: 3812: 3811: 3762: 3761: 3738: 3726: 3716: 3714:complex numbers 3707: 3698: 3683: 3641: 3635:based on your ( 3603: 3580: 3500: 3476:§ Some examples 3472:§ Hilbert space 3461:§ Hilbert space 3434: 3433: 3412: 3411: 3405: 3399: 3301: 3267: 3246:WP:edit warring 3233: 3208: 3207: 3183: 3182: 3171: 3138: 3137: 3115: 3102: 3091: 3063: 3062: 3011: 3010: 2985: 2952: 2848: 2844: 2802: 2801:if and only if 2792: 2775: 2761: 2672: 2671: 2652: 2633: 2631:History section 2569: 2568: 2531: 2530: 2438: 2422: 2417: 2416: 2395: 2385: 2366: 2347: 2328: 2314: 2313: 2292: 2276: 2235: 2234: 2131: 2130: 2062: 2061: 2045: 1998: 1962: 1935: 1934: 1888: 1861: 1860: 1852: 1832: 1822: 1759: 1686: 1625: 1541: 1504: 1503:(Engineering) 1455: 1451: 1372: 1369: 1353:Jakob.scholbach 1330: 1326: 1323: 1321: 1311: 1262: 1247: 1213:—The preceding 1196: 1172: 1166: 1157: 1151: 1140: 1092: 1029: 1028: 964: 919: 907: 881: 880: 851: 835: 783:should move to 652: 643:Oleg Alexandrov 603: 501: 451: 385:+ <y, y: --> 381:+ <y, y: --> 379:+ <y, x: --> 372:+ <y, x: --> 367:+ <y, x: --> 359:+ <y, x: --> 350:- <y, x: --> 342:4 <x, y: --> 335:Now adding the 330:- <y, x: --> 322:- <y, x: --> 321:-(<x, y: --> 315:- <y, x: --> 313:) <1, 1: --> 312:- <y, x: --> 311:-(<x, y: --> 304:- <y, x: --> 302:- <y, x: --> 301:-(<x, y: --> 294:- <y, x: --> 292:- <x, y: --> 284:- <y, x: --> 282:- <y, x: --> 273:- <y, x: --> 271:- <y, x: --> 234:Minkowski space 211: 174: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 3820: 3818: 3810: 3809: 3804: 3799: 3794: 3789: 3784: 3779: 3774: 3764: 3763: 3760: 3759: 3758: 3757: 3723: 3722: 3682: 3679: 3678: 3677: 3676: 3675: 3674: 3673: 3672: 3671: 3670: 3669: 3655: 3614: 3570: 3569: 3568: 3567: 3566: 3565: 3564: 3563: 3562: 3561: 3531: 3524:WP:NOTTEXTBOOK 3485: 3484: 3483: 3468: 3457: 3446: 3442: 3420: 3393: 3389: 3364: 3363: 3362: 3361: 3360: 3357: 3353: 3350: 3343: 3342: 3338: 3337: 3336: 3335: 3331: 3328: 3322: 3321: 3308: 3297: 3296: 3291: 3286: 3280: 3279: 3263: 3262: 3261: 3260: 3242: 3241: 3240: 3229: 3215: 3195: 3191: 3179: 3150: 3146: 3099: 3098: 3087: 3084: 3071: 3059: 3057:to <ul: --> 3054: 3019: 2984: 2981: 2962: 2961: 2951: 2948: 2947: 2946: 2945: 2944: 2943: 2942: 2941: 2940: 2871: 2870: 2867: 2864: 2843: 2840: 2826: 2824: 2823: 2782: 2781: 2763:From article: 2760: 2757: 2756: 2755: 2754: 2753: 2752: 2751: 2731: 2730: 2729: 2728: 2691: 2688: 2685: 2682: 2679: 2651: 2648: 2632: 2629: 2628: 2627: 2626: 2625: 2609: 2608: 2607: 2606: 2600: 2599: 2598: 2597: 2596: 2595: 2594: 2593: 2582: 2579: 2576: 2556: 2553: 2550: 2547: 2544: 2541: 2538: 2521: 2520: 2519: 2518: 2515: 2501: 2500: 2477: 2472: 2467: 2463: 2459: 2456: 2453: 2450: 2445: 2441: 2437: 2434: 2429: 2425: 2402: 2398: 2392: 2388: 2384: 2381: 2378: 2373: 2369: 2365: 2362: 2359: 2354: 2350: 2346: 2343: 2340: 2335: 2331: 2327: 2324: 2321: 2299: 2295: 2291: 2288: 2283: 2279: 2275: 2272: 2269: 2264: 2259: 2256: 2253: 2249: 2245: 2242: 2230: 2229: 2198: 2197: 2196: 2195: 2184: 2181: 2178: 2175: 2172: 2169: 2164: 2161: 2156: 2153: 2150: 2147: 2144: 2141: 2138: 2122: 2121: 2120: 2119: 2108: 2105: 2102: 2099: 2096: 2093: 2090: 2087: 2084: 2081: 2078: 2075: 2072: 2069: 2056: 2055: 2044: 2041: 2040: 2039: 2024: 2023: 2022: 2021: 1986: 1982: 1977: 1971: 1966: 1961: 1958: 1954: 1950: 1946: 1942: 1926: 1925: 1924: 1923: 1912: 1908: 1903: 1897: 1892: 1887: 1884: 1880: 1876: 1872: 1868: 1851: 1848: 1831: 1828: 1827: 1826: 1818: 1815:JohnBlackburne 1791: 1790: 1758: 1755: 1754: 1753: 1752: 1751: 1733: 1732: 1685: 1682: 1681: 1680: 1679: 1678: 1660: 1659: 1624: 1621: 1620: 1619: 1618: 1617: 1616: 1615: 1593: 1592: 1591: 1590: 1573: 1572: 1540: 1537: 1536: 1535: 1491: 1490: 1450: 1447: 1446: 1445: 1444: 1443: 1433:2andrewknyazev 1418: 1417: 1379:131.155.68.160 1368: 1365: 1364: 1363: 1310: 1307: 1306: 1305: 1261: 1258: 1246: 1243: 1242: 1241: 1219:88.111.140.233 1204:= <y,y: --> 1195: 1192: 1191: 1190: 1168: 1162: 1153: 1147: 1139: 1136: 1119: 1118: 1091: 1088: 1054: 1051: 1048: 1045: 1042: 1039: 1036: 1025: 1024: 1023: 1022: 993: 992: 963: 960: 959: 958: 957: 956: 931: 926: 922: 914: 910: 904: 899: 896: 893: 889: 850: 847: 834: 831: 830: 829: 808: 735: 734: 733: 732: 705: 682: 651: 648: 647: 646: 602: 599: 597: 592: 591: 590: 589: 588: 587: 575: 574: 573: 572: 534: 533: 525: 524: 500: 497: 496: 495: 479: 478: 477: 476: 469: 468: 450: 449:Sequilinearity 447: 418: 414: 409: 395: 378:(<x, y: --> 371:(<x, y: --> 365: 358:(<x, y: --> 349:(<x, y: --> 334: 331:)^2 =< 0. 329:(<x, y: --> 327: 326: 325:= 0; therefore 318: 317: 314:(<x, y: --> 308: 307: 298: 297: 291:(<y, x: --> 288: 287: 278: 277: 267: 266: 246: 245: 244: 243: 210: 207: 206: 205: 173: 170: 168: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 3819: 3808: 3805: 3803: 3800: 3798: 3795: 3793: 3790: 3788: 3785: 3783: 3780: 3778: 3775: 3773: 3770: 3769: 3767: 3756: 3752: 3748: 3742: 3737: 3736: 3735: 3734: 3733: 3732: 3729: 3719: 3715: 3710: 3706: 3701: 3696: 3692: 3688: 3687: 3686: 3680: 3668: 3664: 3660: 3656: 3653: 3649: 3648: 3647: 3644: 3638: 3634: 3630: 3629: 3628: 3624: 3620: 3615: 3611: 3610: 3609: 3606: 3600: 3596: 3592: 3588: 3587: 3586: 3583: 3576: 3572: 3571: 3560: 3556: 3552: 3547: 3546: 3545: 3541: 3537: 3532: 3529: 3525: 3521: 3520: 3519: 3515: 3511: 3504: 3499: 3498: 3497: 3493: 3489: 3486: 3481: 3477: 3473: 3469: 3466: 3462: 3458: 3444: 3408: 3402: 3398:I agree that 3397: 3396: 3394: 3390: 3387: 3384: 3380: 3379: 3378: 3374: 3370: 3365: 3358: 3354: 3351: 3347: 3346: 3345: 3344: 3340: 3339: 3332: 3329: 3326: 3325: 3324: 3323: 3319: 3318: 3316: 3313: 3309: 3307: 3304: 3299: 3298: 3295: 3292: 3290: 3287: 3285: 3282: 3281: 3277: 3271: 3265: 3264: 3259: 3255: 3251: 3247: 3243: 3239: 3236: 3230: 3213: 3193: 3180: 3177: 3174: 3168: 3164: 3163: 3148: 3135: 3131: 3127: 3123: 3122: 3121: 3118: 3112: 3106: 3101: 3100: 3097: 3094: 3088: 3085: 3060: 3055: 3052: 3051: 3050: 3049: 3045: 3041: 3037: 3032: 3008: 3004: 3000: 2995: 2993: 2989: 2982: 2980: 2979: 2975: 2971: 2967: 2960: 2957: 2956: 2955: 2949: 2939: 2936: 2932: 2931: 2930: 2926: 2922: 2918: 2917: 2916: 2913: 2909: 2905: 2904: 2903: 2899: 2895: 2891: 2887: 2883: 2882: 2881: 2880: 2877: 2868: 2865: 2862: 2861: 2860: 2858: 2852: 2841: 2839: 2838: 2834: 2830: 2829:62.77.218.240 2822: 2818: 2814: 2810: 2805: 2800: 2795: 2790: 2787: 2786: 2785: 2778: 2773: 2769: 2766: 2765: 2764: 2758: 2750: 2746: 2742: 2737: 2736: 2735: 2734: 2733: 2732: 2727: 2723: 2719: 2715: 2714: 2713: 2709: 2705: 2686: 2683: 2680: 2669: 2668: 2667: 2666: 2662: 2658: 2649: 2647: 2646: 2642: 2638: 2630: 2624: 2620: 2616: 2613: 2612: 2611: 2610: 2604: 2603: 2602: 2601: 2580: 2577: 2574: 2554: 2551: 2545: 2542: 2539: 2529: 2528: 2527: 2526: 2525: 2524: 2523: 2522: 2516: 2513: 2509: 2505: 2504: 2503: 2502: 2499: 2495: 2491: 2475: 2470: 2465: 2461: 2457: 2451: 2448: 2443: 2439: 2432: 2427: 2423: 2390: 2386: 2382: 2379: 2376: 2371: 2367: 2360: 2352: 2348: 2344: 2341: 2338: 2333: 2329: 2322: 2319: 2297: 2293: 2289: 2281: 2277: 2273: 2270: 2262: 2257: 2254: 2251: 2243: 2240: 2232: 2231: 2228: 2224: 2220: 2216: 2215: 2214: 2213: 2209: 2205: 2201: 2182: 2176: 2173: 2170: 2159: 2154: 2148: 2145: 2142: 2139: 2129: 2128: 2127: 2126: 2125: 2106: 2100: 2097: 2094: 2088: 2085: 2079: 2076: 2073: 2070: 2060: 2059: 2058: 2057: 2053: 2050: 2049: 2048: 2042: 2038: 2034: 2030: 2026: 2025: 2018: 2014: 2010: 2006: 2002: 1984: 1969: 1959: 1948: 1933: 1932: 1931: 1930: 1929: 1910: 1895: 1885: 1874: 1859: 1858: 1857: 1856: 1855: 1849: 1847: 1846: 1842: 1838: 1829: 1825: 1821: 1816: 1811: 1807: 1806: 1805: 1804: 1800: 1796: 1789: 1785: 1781: 1776: 1775: 1774: 1773: 1769: 1765: 1756: 1750: 1746: 1742: 1737: 1736: 1735: 1734: 1731: 1727: 1723: 1718: 1717: 1716: 1715: 1711: 1707: 1703: 1701: 1698:For instance 1696: 1694: 1689: 1677: 1673: 1669: 1664: 1663: 1662: 1661: 1658: 1654: 1650: 1645: 1644: 1643: 1642: 1638: 1634: 1628: 1622: 1614: 1611: 1609: 1608: 1603: 1599: 1598: 1597: 1596: 1595: 1594: 1589: 1585: 1581: 1577: 1576: 1575: 1574: 1571: 1567: 1563: 1558: 1557: 1556: 1555: 1551: 1547: 1538: 1534: 1531: 1529: 1524: 1523: 1522: 1520: 1516: 1512: 1508: 1500: 1497: 1496:Inner Product 1489: 1486: 1484: 1480: 1475: 1474: 1473: 1471: 1467: 1463: 1462:79.246.47.200 1459: 1448: 1442: 1438: 1434: 1430: 1426: 1422: 1421: 1420: 1419: 1416: 1412: 1408: 1404: 1400: 1396: 1392: 1391: 1390: 1388: 1384: 1380: 1376: 1366: 1362: 1358: 1354: 1350: 1346: 1345: 1344: 1343: 1334: 1329: 1319: 1318:Google Chrome 1315: 1309:HTML problems 1308: 1304: 1300: 1296: 1292: 1288: 1287: 1282: 1281:inner product 1278: 1277: 1276: 1275: 1271: 1267: 1260:Map symbolism 1259: 1257: 1256: 1253: 1244: 1240: 1237: 1233: 1232: 1231: 1228: 1224: 1220: 1216: 1210: 1193: 1189: 1186: 1182: 1181: 1180: 1179: 1176: 1171: 1165: 1161: 1156: 1150: 1146: 1137: 1135: 1133: 1129: 1125: 1117: 1114: 1110: 1105: 1104: 1103: 1101: 1098: 1089: 1087: 1086: 1083: 1082:Functor salad 1079: 1076: 1073: 1071: 1066: 1049: 1046: 1043: 1034: 1021: 1018: 1014: 1013: 1012: 1009: 1004: 1003: 1002: 1001: 998: 991: 989: 985: 979: 978: 972: 971:The map from 969: 968: 967: 961: 955: 951: 947: 924: 920: 912: 908: 902: 897: 894: 891: 887: 878: 874: 873: 871: 870: 869: 868: 864: 860: 856: 848: 846: 845: 841: 840: 832: 828: 824: 820: 815: 811: 810: 809: 806: 805: 802: 798: 794: 790: 789:Inner Product 786: 785:inner product 782: 778: 777:inner product 774: 770: 769:inner product 766: 765:inner product 762: 758: 754: 751: 747: 746:inner product 742: 740: 731: 728: 724: 721:I agree with 720: 719: 718: 714: 710: 706: 703: 699: 695: 691: 687: 686:inner product 683: 680: 676: 675: 674: 673: 670: 665: 664:Inner product 661: 657: 649: 644: 640: 636: 635: 634: 632: 627: 625: 621: 617: 613: 609: 600: 598: 595: 585: 581: 580: 579: 578: 577: 576: 570: 566: 562: 558: 554: 550: 546: 542: 538: 537: 536: 535: 531: 527: 526: 522: 518: 514: 510: 509: 508: 505: 498: 494: 490: 486: 481: 480: 473: 472: 471: 470: 466: 461: 460: 459: 457: 448: 446: 444: 440: 436: 432: 427: 421: 419: 416: 412: 407: 404: 398: 393: 387: 383:<y, y: --> 376: 374:<y, y: --> 369: 363: 361:<y, y: --> 356: 354:<y, y: --> 352:<y, x: --> 347: 345:<y, y: --> 343:<y, x: --> 340: 338: 332: 320: 319: 310: 309: 300: 299: 290: 289: 280: 279: 269: 268: 263:<1, 1: --> 262: 261: 260: 257: 255: 251: 242: 239: 235: 231: 230:bilinear form 227: 226: 224: 223:Michael Hardy 220: 219:inner product 216: 215: 214: 208: 204: 200: 196: 191: 190: 189: 188: 184: 180: 172:Ordered Field 171: 169: 153: 149: 148:High-priority 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 73:High‑priority 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 3724: 3717: 3708: 3705:real numbers 3699: 3684: 3528:WP:SYNTHESIS 3406: 3400: 3383:§ Definition 3033: 3007:WP:SYNTHESIS 3003:WP:EDITORIAL 2996: 2992:§ Definition 2986: 2965: 2963: 2958: 2953: 2889: 2885: 2872: 2856: 2850: 2845: 2825: 2803: 2793: 2788: 2783: 2776: 2767: 2762: 2741:50.125.86.70 2653: 2634: 2202: 2199: 2123: 2046: 1999:— Preceding 1927: 1853: 1833: 1792: 1760: 1741:50.35.97.238 1704: 1697: 1692: 1690: 1687: 1629: 1626: 1606: 1542: 1511:79.246.54.99 1505:— Preceding 1501: 1492: 1478: 1456:— Preceding 1452: 1428: 1424: 1398: 1394: 1370: 1314:This section 1312: 1284: 1263: 1248: 1211: 1197: 1169: 1163: 1159: 1154: 1148: 1144: 1141: 1120: 1093: 1080: 1077: 1074: 1069: 1067: 1026: 994: 981: 974: 970: 965: 854: 852: 842: 836: 819:Jitse Niesen 813: 807: 764: 760: 756: 752: 749: 745: 743: 736: 723:Jitse Niesen 709:Jitse Niesen 689: 685: 653: 628: 623: 619: 615: 611: 607: 604: 596: 593: 569:Jitse Niesen 564: 560: 556: 552: 548: 544: 540: 516: 506: 502: 485:50.35.97.238 452: 422: 408: 405: 399: 394: 388: 377: 370: 364: 357: 348: 341: 336: 333: 328: 258: 247: 218: 212: 195:50.35.97.238 179:50.35.97.238 175: 167: 147: 107: 51:WikiProjects 34: 3747:Danstronger 3659:Danstronger 3637:Danstronger 3619:Danstronger 3595:Danstronger 3551:Danstronger 3510:Danstronger 3369:Danstronger 3341:Substantive 2842:POV article 2718:CountMacula 2702:is real. -- 2657:CountMacula 2637:CountMacula 1810:dot product 1795:Bxb Grxmmxn 1405:article. -- 1373:—Preceding 1134:gsspradlin 988:injectivity 877:dot product 814:dot product 793:Dot Product 757:dot product 750:dot product 702:dot product 694:dot product 690:dot product 660:dot product 435:82.13.19.79 429:—Preceding 123:Mathematics 114:mathematics 70:Mathematics 3766:Categories 3652:WP:MOSBOLD 3465:§ Examples 1837:Gsspradlin 1602:isomorphic 1479:Definition 1124:Gsspradlin 977:dual space 859:Salix alba 528:Fixed. -- 397:; thereby 3274:violated 2809:countable 2799:separable 2772:separable 2052:Linearity 1764:Spirit469 1286:TedPavlic 1097:Sick0Fant 962:What map? 946:James pic 584:Walt Pohl 530:Walt Pohl 521:Walt Pohl 499:Bilinear? 410:Regards, 39:is rated 3697:denoted 3599:D.Lazard 3536:D.Lazard 3503:D.Lazard 3488:D.Lazard 3320:Cosmetic 3270:D.Lazard 3250:D.Lazard 3206:" with " 3132:. Also, 3105:D.Lazard 3040:D.Lazard 2999:§ Remark 2789:Theorem. 2768:Theorem. 2013:contribs 2001:unsigned 1668:Gwideman 1633:Gwideman 1580:Gwideman 1546:Gwideman 1507:unsigned 1458:unsigned 1427:here. A 1375:unsigned 1252:Ustad NY 1215:unsigned 631:User:--- 551:and not 465:FunnyMan 431:unsigned 3741:Mgkrupa 3727:Mgkrupa 3695:scalars 3642:Mgkrupa 3604:Mgkrupa 3581:Mgkrupa 3463:inside 3302:Mgkrupa 3234:Mgkrupa 3172:Mgkrupa 3134:MOS:VAR 3116:Mgkrupa 3092:Mgkrupa 2988:Mgkrupa 2970:YohanN7 2935:Quondum 2921:YohanN7 2912:Quondum 2908:physics 2894:YohanN7 2876:Quondum 2813:YohanN7 2704:Zundark 1722:Zundark 1695:field? 1607:Quondum 1528:Quondum 1483:Quondum 1407:Zundark 1399:induced 1295:contrib 1175:Matumba 1095:care?-- 975:to the 839:Tbsmith 519:. -- 417:W ~@) R 238:Rgdboer 150:on the 41:B-class 3126:WP:BRD 3036:WP:BRD 2807:has a 1481:). — 1425:metric 1395:metric 1327:Dr Dec 1266:Unfree 622:where 415:rank 305:): --> 295:): --> 285:): --> 274:): --> 177:field. 47:scale. 3691:field 2508:LutzL 2506:Dear 2490:LutzL 2219:LutzL 1820:deeds 1780:LutzL 1236:CSTAR 1017:lethe 1008:lethe 997:CSTAR 727:CSTAR 456:CSTAR 392:))^2 324:: --> 275:: --> 264:: --> 228:Yes, 28:This 3751:talk 3663:talk 3623:talk 3555:talk 3540:talk 3526:and 3514:talk 3492:talk 3432:and 3404:and 3373:talk 3314:and 3254:talk 3124:Per 3044:talk 3005:and 2974:talk 2925:talk 2898:talk 2833:talk 2817:talk 2770:Any 2745:talk 2722:talk 2708:talk 2661:talk 2641:talk 2619:talk 2512:talk 2494:talk 2223:talk 2208:talk 2033:talk 2029:Bdmy 2009:talk 1841:talk 1799:talk 1784:talk 1768:talk 1745:talk 1726:talk 1710:talk 1672:talk 1653:talk 1637:talk 1584:talk 1566:talk 1550:talk 1515:talk 1466:talk 1437:talk 1429:norm 1411:talk 1383:talk 1357:talk 1333:Talk 1291:talk 1270:talk 1223:talk 1128:talk 1113:Talk 950:talk 863:talk 823:talk 748:and 713:talk 688:and 658:and 489:talk 439:talk 403:), 276:= 0, 199:talk 183:talk 142:High 3693:of 3597:or 3474:in 2797:is 1693:any 1649:JBL 1562:JBL 1324:~~ 1109:MFH 761:not 759:is 753:are 725:.-- 610:× 375:), 316:) = 3768:: 3753:) 3665:) 3625:) 3601:? 3557:) 3542:) 3516:) 3494:) 3375:) 3256:) 3228:"? 3178:). 3046:) 2976:) 2968:. 2927:) 2900:) 2892:. 2835:) 2819:) 2747:) 2724:) 2710:) 2690:⟩ 2678:⟨ 2663:) 2643:) 2621:) 2581:0. 2552:≥ 2549:⟩ 2537:⟨ 2496:) 2471:∗ 2455:⟩ 2436:⟨ 2401:⊤ 2380:… 2361:⋅ 2342:… 2290:⋅ 2287:⟩ 2268:⟨ 2248:∑ 2225:) 2210:) 2180:⟩ 2168:⟨ 2163:¯ 2152:⟩ 2137:⟨ 2104:⟩ 2092:⟨ 2083:⟩ 2068:⟨ 2035:) 2015:) 2011:• 1970:∗ 1960::= 1957:⟩ 1941:⟨ 1896:∗ 1886::= 1883:⟩ 1867:⟨ 1843:) 1801:) 1786:) 1770:) 1747:) 1728:) 1720:-- 1712:) 1674:) 1655:) 1647:-- 1639:) 1586:) 1568:) 1552:) 1517:) 1468:) 1439:) 1413:) 1385:) 1359:) 1337:~~ 1301:) 1272:) 1225:) 1130:) 1102:Φ 1053:⟩ 1044:⋅ 1041:⟨ 1038:↦ 952:) 930:¯ 888:∑ 865:) 825:) 715:) 641:. 624:V* 618:× 608:V* 559:→ 555:× 547:→ 543:× 491:) 441:) 386:, 362:. 355:, 346:: 339:: 265:0. 256:: 201:) 185:) 3749:( 3743:: 3739:@ 3721:. 3718:C 3709:R 3700:F 3661:( 3621:( 3553:( 3538:( 3512:( 3505:: 3501:@ 3490:( 3482:. 3467:. 3445:. 3441:C 3419:R 3407:C 3401:R 3371:( 3311:( 3272:: 3268:@ 3252:( 3214:F 3194:, 3190:F 3149:, 3145:F 3107:: 3103:@ 3070:F 3061:" 3042:( 3018:F 2972:( 2923:( 2896:( 2855:§ 2853:. 2831:( 2815:( 2804:H 2794:H 2777:V 2743:( 2720:( 2706:( 2687:x 2684:, 2681:x 2659:( 2639:( 2617:( 2578:= 2575:x 2555:0 2546:x 2543:, 2540:x 2510:( 2492:( 2476:v 2466:k 2462:e 2458:= 2452:v 2449:, 2444:k 2440:e 2433:= 2428:k 2424:x 2397:) 2391:n 2387:x 2383:, 2377:, 2372:1 2368:x 2364:( 2358:) 2353:n 2349:e 2345:, 2339:, 2334:1 2330:e 2326:( 2323:= 2320:v 2298:i 2294:e 2282:i 2278:e 2274:, 2271:v 2263:n 2258:1 2255:= 2252:k 2244:= 2241:v 2221:( 2206:( 2183:. 2177:y 2174:, 2171:x 2160:a 2155:= 2149:y 2146:, 2143:x 2140:a 2107:. 2101:y 2098:, 2095:x 2089:a 2086:= 2080:y 2077:, 2074:x 2071:a 2031:( 2007:( 1985:. 1981:x 1976:M 1965:y 1953:y 1949:, 1945:x 1911:. 1907:y 1902:M 1891:x 1879:y 1875:, 1871:x 1839:( 1797:( 1782:( 1766:( 1743:( 1724:( 1708:( 1670:( 1651:( 1635:( 1582:( 1564:( 1548:( 1513:( 1477:( 1464:( 1435:( 1409:( 1381:( 1355:( 1335:) 1331:( 1299:@ 1297:/ 1293:/ 1289:( 1268:( 1229:. 1221:( 1170:k 1167:} 1164:k 1160:e 1155:k 1152:} 1149:k 1145:e 1143:{ 1126:( 1111:: 1070:V 1050:x 1047:, 1035:x 1006:- 990:: 980:V 973:V 948:( 925:n 921:y 913:n 909:x 903:n 898:0 895:= 892:i 861:( 821:( 711:( 704:. 681:? 620:V 616:V 612:V 565:C 561:R 557:V 553:V 549:F 545:V 541:V 517:R 487:( 463:- 437:( 413:F 306:= 296:= 286:= 197:( 181:( 154:. 53::

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