Talk:Cauchy–Schwarz inequality

Big update 2016-5-20

Performed many updates to remove the "need more references" and "expert needed" alert boxes, including

• added more inline references
• organized the proof section to have two different proofs and links to more proofs. I don't think there is anything to gain by having more proofs. Also, I swapped out one proof as the 2nd one contains an error/is unclear (see the section "Questions about the Alternative Proof" below).
• Special cases section: cleaned it up to use the same <u,v> notation as in the rest of the page. Also removed one proof but kept one proof that works for the real case only as it might be easier for someone in high school to understand versus having to know about inner products.
• Applications section: organized it.
• Generalizations section: the content is super advance for this page. I removed the proofs and make the theorems the key points.
• fixed some other ref/link issues

DrWikiWikiShuttle (talk) 19:34, 20 May 2016 (UTC)

Untitled

Wallis' name deserves to go on this. I did not confuse it with the infinite product which converges to pi/4. This name appears in Montgomery and Vaughan's book "Multiplicative Number Theory 1".

Proof

A proof was requested by an anon

A proof appears in the article inner product space. A moment ago, I added that same proof to this page. Michael Hardy 22:34, 11 Nov 2004 (UTC)

An alternate proof, which i learnt is as follows: cos x = u . v / ||u|| ||v|| as cos x is between -1 and 1, the absolute value of the denomenator must be larger or equal to the numerator, hence u . v <= ||u|| ||v|| TheDarkLeaf 17:30, 19 June 2005 (AEST)

But first you need to establish that the cosine does play that role. You can give an easy intuitive geometric argument, but whether it works in, e.g., infinite-dimensional spaces may be dubious. Michael Hardy 23:24, 19 Jun 2005 (UTC)

While it is likely that the only people who might be interested in this page would already be familiar with the various mathmatical symbols used, it wouldn't hurt to add a little to the end or somehow link them to their explaination page (dunno what they are or how to do that). -FjordPrefect

There are conspicuous links to inner product space and related articles; that's where those explanations should be sought. Michael Hardy 23:58, 12 August 2005 (UTC)

How can you justify allowing lambda to be <x,y>*<y,y>^-1? I just do not see how this is a general proof. —Preceding unsigned comment added by 216.106.49.131 (talk) 19:00, 23 November 2007 (UTC)

The expression has been shown to be valid for all complex $\lambda$, and $y$ is assumed non-zero, so obviously we can take $\lambda =\langle x,y\rangle \cdot \langle y,y\rangle ^{-1}$. --Zundark (talk) 19:24, 23 November 2007 (UTC)

I corrected a little error in the proof :

$\langle x-\lambda y|x-\lambda y\rangle =\langle x|x\rangle |\lambda |^{2}\langle y|y\rangle -\lambda ^{*}\langle y|x\rangle -\lambda \langle x|y\rangle$

That's the correct order the complex conjugate of lambda coming from the ket part. Hence the lambda that allows us to conclude is the complex conjugate of the one that was presented :

$\lambda =\langle y,x\rangle \cdot \langle y,y\rangle ^{-1}$

Indeed we then get :

$\langle x-\lambda y|x-\lambda y\rangle =\langle x|x\rangle |\langle y|x\rangle |^{2}\langle y|y\rangle ^{-1}-|\langle x|y\rangle |^{2}\langle y|y\rangle ^{-1}-\langle y|x\rangle \langle x|y\rangle \langle y|y\rangle ^{-1}$

The second and the last term cancel nicely and we're done.

ThibautLienart (talk) 15:12, 3 June 2009 (UTC)

But the proof doesn't use bra-ket notation, it uses the usual mathematical notation, and therefore conforms to the usual mathematical convention (linearity in the first argument of the inner product). So it was correct as it was, and is now wrong. --Zundark (talk) 18:35, 3 June 2009 (UTC)

My bad, I had just been working on the proof with the ket-notation and it's the "opposite", my apologies (reverted to the previous version). ThibautLienart (talk) 13:53, 5 June 2009 (UTC)

TOD: Specific Cases

As promised in the first sentence we need to add specific information on how Cauchy-Schwartz applies to:

• Infinite series
• Integration
• Variances / covariances. Perhaps a link to the Cramér–Rao bound whose proof relies on Cauchy-Schwartz.

Eug (talk) 03:41, 15 April 2008 (UTC)

another proof for cauchy-schwarz inequality

for two non zero vectors x,y in eucledian space V. the angle Q(x,y) formed by x and y is defined by-

cosQ(x,y) = <x,y>/(||x||.||y||) →(1)

as we know that-

-1 ≤ cosQ ≤1

we can say-

0 ≤ |cosQ| ≤ 1

that is cosQ is always less than 1. using this in expression→(1) we get

1 ≥ |<x,y>|/(||x||.||y||)

=> ||x||.||y|| ≥ |<x,y>|

=> |<x,y>| ≤ ||x||.||y|| Abhishchauhan (talk) 08:31, 20 September 2008 (UTC) —Preceding unsigned comment added by Abhishchauhan (talk • contribs) 08:07, 20 September 2008 (UTC)

I know it's poor form to edit other people's comments, but I fixed the formatting in the above comment because it was making this page wide enough for a horizontal scroll bar appear. The mathematical response can be found above where someone suggested the same thing: normally we *define* angle using the above relation involving cosQ, so we need to prove Cauchy-Schwarz without it otherwise we have a circular argument. Quietbritishjim (talk) 09:46, 19 July 2012 (UTC)

Common misspelling

The page currently seems to claim that Herman Schwarz's name is commonly misspelled as Schwartz. There's no such claim on his own wikipedia page. Did this pages author mean that the inequality is frequently misattributed to Laurent Schwartz?

I find it hard to imagine anyone attributing this inequality to Laurent Schwartz. Laurent Schwartz's work was done in the middle of the 20th century, if I'm not mistaken, whereas this inequality seems obviously much older.

It seems possible that the name is not frequently misspelled when one writes about the person, since writing about the person focuses one's attention on such things as what his name is, but is frequently misspelled when one thinks about this inequality. Notice how often people who write Wikipedia articles write "Xmith's theorem states that blah blah blah blah...." without mentioning who Xmith is—they're not thinking about the person at all. Michael Hardy (talk) 15:38, 20 July 2009 (UTC)

Agreed. People remember the name as they heard it in countless math lectures and try to spell it how it sounds. There are countless good references that spell the name wrong in text (for example, Bertsekas and Tsitsikis 1997 often-referenced book Parallel and Distributed Computation: Numerical Methods, which introduces the "Schwartz inequality" on page 621 in its appendix on linear algebra). It's possibly the fault of math instructors for not spelling it out at least the first few times they say it. —TedPavlic (talk/contrib/@) 12:02, 21 July 2009 (UTC)

common name

Is there a reference for the most commonly used name of the inequality?? since it was discovered by Cauchy and Bunyakovsky , it seems that Schwarz had very little to do with it. Is the "Cauchy–Schwarz" name a historical artefact? Where did it come from, any sources? 58.247.201.245 (talk) 12:23, 4 September 2009 (UTC)

I think the article should be renamed Cauchy-Bunyakovsky-Schwarz. — Preceding unsigned comment added by 130.229.155.63 (talk) 14:50, 4 November 2011 (UTC)

In france, mathematicians calls it "The Cauchy Inequality", in germany "The Schwarz Inequality" and in russia... well you get the point. Most mathematical discoveries was made independantly by several mathematicians, most often whilst trying to prove something else. This naming problem can be discussed when it comes to almost any mathematical statement. It´s not relevant what you call it, it´s the content of the theorem.

It is commonly called the "Schwartz Inequality" in British mathematical texts. Since British mathematicians were at the coal face during these times, I'd put more faith in their description. "Stefan's Law" is similar. It becomes the "Stefan-Boltzmann Law" in American texts. A law should be named after the person to understand the true broader meaning. For example Kelvin notes that "everywhere the flow is parallel to the lines of constant pressure" when he completed the derivation of the Long Term Tides problem. However, the discovery and fuller understanding of geostrophy is attributed to Gustav Rossby. The broader implication is that most of the ocean's energy is locked up as potential energy. Kelvin did not understand that.203.87.98.101 (talk) 01:13, 8 January 2016 (UTC)

I'm re-adding the name Bunyakovsky because there is enough evidence that "Cauchy-Bunyakovsky-Schwarz inequality" is in widespread use. I Google'd "Cauchy-Bunyakovsky-Schwarz inequality" (exact match) and got 10 pages worth of results. Googl'ing for "Cauchy-Schwarz inequality" brought in 11 pages worth of results. Here are three reliable sources, and I am sure there are more:

1 (lecture notes published by a Math Prof.)

2 (lecture notes published by a Math & Computer Science Prof.)

3 (H.A.Schwarz biography sketch published by School of Math & Statistics at University of St. Andrews, Scotland)

StrokeOfMidnight (talk) 23:01, 10 August 2021 (UTC)

generalization

I do not have the time to go into this right now, but a really important generalization which should be mentioned in the Generalizations section is the Holder inequality. --Zvika (talk) 16:18, 19 January 2010 (UTC)

One could do worse here than to quote (briefly) and then paraphrase, from chapter 9 of "The Cauchy-Schwarz Master Class" (J. Michael Steele, Cambridge 2004). — Preceding unsigned comment added by 74.192.201.205 (talk) 19:36, 30 October 2013 (UTC)

An intuitive proof

Here's another proof for Cauchy-Schwarz inequality, which I think is much more intuitive than the current one. First deal with the case $x=0$, which is trivial. Then assume $x\neq 0$, in which case the CS-inequality is equivalent to $\lVert y_{\parallel }\rVert \leq \lVert y\rVert$, where $y_{\parallel }={\frac {\langle x,y\rangle }{\lVert x\rVert }}{\frac {x}{\lVert x\rVert }}$ is the orthogonal projection of $y$ to $x$. That is, the CS-inequality states that the length of a vector does not grow in an orthogonal projection to a vector. Guided by this intuition, decompose $y=y_{\perp } y_{\parallel }$, where $y_{\perp }=y-y_{\parallel }$. By construction $\langle y_{\perp },y_{\parallel }\rangle =0$. Now $\langle y,y\rangle =\langle y_{\perp },y_{\perp }\rangle \langle y_{\parallel },y_{\parallel }\rangle 2\langle y_{\perp },y_{\parallel }\rangle =\langle y_{\perp },y_{\perp }\rangle \langle y_{\parallel },y_{\parallel }\rangle \geq \langle y_{\parallel },y_{\parallel }\rangle$, proving the statement. What do you think? --Kaba3 (talk) 10:40, 19 May 2010 (UTC)

I like your proof. Bear in mind that the standard proof is a cute example of the amplification trick. —Preceding unsigned comment added by Tercer (talk • contribs) 04:24, 28 May 2010 (UTC)

User:Chinju has put a proof in similar lines. I think this one looks better. Also, I like the standard proof that we had before better. What do you guys think? --Memming (talk) 20:46, 16 August 2011 (UTC)

No definition of 2-positivity in the article

There's no definition of 2-positivity given. This would be nice, but I don't know what 2-positivity is so I can't put it in myself. Will anyone do this? —Preceding unsigned comment added by 163.1.180.170 (talk) 17:47, 25 October 2010 (UTC)

I've no idea either, but I've put in a request at the Talk page of the editor who added this section, although this editor has not edited for 3 months. If there's no response within a month or so I suggest we delete the section for the moment. Qwfp (talk) 18:59, 25 October 2010 (UTC)

2-postive is defined at Choi's theorem on completely positive maps, which I'm sure is the link intended for positive map which I've updated. I can't make sense of it having read that page, so can't confirm if it's right, and unless a source can be found or someone steps in to make something of it I'd say delete it too.--JohnBlackburne^words_deeds 19:47, 25 October 2010 (UTC)

See also

• A generalization of Cauchy - Schwarz inequality(Van khea (talk) 15:34, 15 December 2010 (UTC))
• Example of Cauchy–Schwarz inequality usage for linear independent vectors Interactive program and tutorial.
• The Cauchy-Schwarz Master Class, J. Michael Steele, Cambridge University Press, 2004. — Preceding unsigned comment added by 74.192.201.205 (talk) 19:28, 30 October 2013 (UTC)

Disputed alternative name "Shivani-Hesha inequality"

There is currently an edit dispute going on regarding the addition of the name "Shivani-Hesha inequality" to the start of the lead section. The reference given is: The Applied Mathamatics for Class XII, CBSE BOARD OF EDUCATION, NEW DELHI, INDIA.page no.23, chapter "Complex Inequality"

A reminder of the facts: Names given in Wikipedia articles (if not just descriptive titles) must be names that are already in common use, not just a name that you think *ought* to be used. In mathematics theorems are often named after the "wrong" person, and it's not up to Wikipedia to "correct" them. What's more, like everything on Wikipedia, that a name is in common use must be verifiable i.e. checkable on reliable third-party sources.

My opinion is that the cited source is in no way enough for the claim that "Shivani-Hesha inequality" is in common use. This inequality is so widespread that any common name for it would have a multitude of sources. Imagine someone disputed that it's called "Cauchy-Schwarz"... we'd have a flood of reliable references to give them! But the only source given for this name is an obscure government document that I can't get hold of. Is there even one published academic textbook that uses the name? A Google search only gives this article, while Google Books and Google Scholar give no results at all. Searching without quotes only adds irrelevant results to the Google search, one result to the books search (I can't check it, but it doesn't look relevant) and none to the Scholar search. I know that Google is not normally sufficient to judge whether a name is in common use, but given that similar searches for "Cauchy-Schwarz inequality" give countless results, there should be at least *some* results if this really were in use.

I'm removing the name until reliable sources are given showing that the name is in common use. Quietbritishjim (talk) 10:14, 19 July 2012 (UTC)

It is simply the Schwartz Inequality in British texts, and has been for a long time.203.87.98.101 (talk) 01:29, 8 January 2016 (UTC)

It is simply the Cauchy–Bunyakovsky inequality in Russian texts, and has been for a long time. :) 81.20.68.181 (talk) 11:47, 24 March 2016 (UTC)

There is also a section about a "Sedrakyan" inequality, which is equally obscure, given that it is actually Bergstrom's inequality, see https://www.cut-the-knot.org/m/Algebra/BergstromInequality.shtml (at google, it wins out by 80K to 6K links)— Preceding unsigned comment added by 2A02:586:4B25:8DC6:50DC:FA83:296C:3EE8 (talk) 22:11, 7 September 2023 (UTC)

Reforming?

There is currently a section titled "Reforming Cauchy-Schwarz Inequality for Cross Product." I can't even parse this. Is "reforming" supposed to be a verb? The section title sounds like some kind of garbled version of the title of a research paper. Jzimba (talk) 21:07, 1 April 2014 (UTC)

As a mathematical physicist, I used the Cauchy-Schwarz inequality constantly in my work, taught it to undergraduates, and so on. I do not recall ever having the occasion to formulate a homologue of the inequality for the vector cross product in three-dimensional space. The Cauchy-Schwarz inequality is always understood as a statement about inner products. The vector product is not an inner product. The section makes little sense here, and I deleted it. One could imagine restoring it, provided the relevance of the result to the (actual) Cauchy-Schwarz inequality and its overall importance were better established; but frankly I doubt that would be a promising undertaking. Jzimba (talk) 16:08, 2 April 2015 (UTC)

This whole article seems to have been written by QM physicists. Complex numbers are phasors (no physicist knows that) yet the inequality still applies. It is a general property of algebras, since inner products can be broadly defined. The vector cross product cannot be generally defined, and does not exist for complex numbers. (The classic undergraduate 2*Pi*C circulation problem in fluid dynamics can be trivially solved with the residue theorem since no use is made of the cross product.)203.87.98.101 (talk) 01:32, 8 January 2016 (UTC)

Questions about the "Alternative Proof"

I believe alternative proof is incorrect: the main inequality has $2{\text{Re}}\left(\left\langle {\frac {\lambda v}{\|v\|}},{\frac {u}{\|u\|}}\right\rangle \right)$ instead of $2{\text{Re}}\left(\left\langle {\frac {\lambda u}{\|u\|}},{\frac {v}{\|v\|}}\right\rangle \right)$ or equivalently $2{\text{Re}}\left(\left\langle {\frac {v}{\|v\|}},{\frac {\lambda u}{\|u\|}}\right\rangle \right)$. If this is fixed then it seems to fall apart. 146.115.186.20 (talk) 04:03, 10 January 2016 (UTC)

In the "Alternative proof" the statement finishes with:

$|\langle u,v\rangle |={\text{Re}}\left(\lambda \langle u,v\rangle \right)\leq \|u\|\|v\|$.

However I disagree that

$|\langle u,v\rangle |={\text{Re}}\left(\lambda \langle u,v\rangle \right)$,

because $\lambda$ can be either negative or positive which would change the inequality. Perhaps you can enlighten me as to the reasoning behind neglecting $\lambda$ in the final step.

Physics

This is just a placeholder comment to note that the Physics section is very minimal. The inequality is used often where math is used in physics, so it should not be hard to flesh out the section with more examples. Jzimba (talk) 16:38, 2 April 2015 (UTC)

The Heisenberg Uncertainty Principle (via the Robertson–Schrödinger uncertainty relations) is an application of the Cauchy-Schwarz inequality. However, the See Also link to the HUP was removed. Maybe an expanded Physics application section could give more context and help protect such examples. Simon 10:19, 5 March 2017 (UTC)

Notation is not uniform

The statement of the inequality denotes the vectors by x and y. The proof of the inequality denotes the vectors by u and v. In a reference written by a single author this would never, of course, happen. I would recommend that the inequality and its proofs all use the same notation for the vectors. Jzimba (talk) 17:50, 14 April 2015 (UTC)

Assessment comment

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

Please add useful comments here. --Cronholm144 21:07, 21 May 2007 (UTC) Note: These ratings are not set in stone, please change them as the article progresses. the article contains the statement, "One can see from the proof that the Cauchy–Schwarz inequality is a consequence of the positivity and anti-symmetry inner-product axioms." This is incorrect. The theorem is also true if one only has a semi-inner product to hand. the proof is slightly more detailed, but positivity is not necessary. Al1ex3 (talk) 16:04, 22 April 2009 (UTC)

Last edited at 16:04, 22 April 2009 (UTC). Substituted at 19:51, 1 May 2016 (UTC)

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Removing statement from introduction that Holder generalizes Cauchy-Schwarz

I removed the statement "It has a number of generalizations, among them Hölder's inequality." from the opening paragraph of the article. Here is the edit.

It is not generally true that Holder generalizes Cauchy-Schwarz and the statement is quite misleading. Cauchy-Schwarz is an inequality on arbitrary inner product spaces; Holder's inequality applies to Lp normed vector spaces. For all p not equal to 2, Lp spaces cannot be said to be inner product spaces, so there is no notion of a "Cauchy-Schwarz" inequality for general Lp. Conversely, the CS inequality can apply to inner product spaces which have nothing to do with Lp, where there is no notion of a Holder inequality.

The original statement, which only holds for L2, is itself not even accurate, since Holder's and CS are actually equivalent, see e.g. Finol, Carlos, and Marek Wójtowicz. "Cauchy-Schwarz and Hölder’s inequalities are equivalent." Divulgaciones Matemáticas 15.2 (2007): 143-147. APA .

Bulkroosh (talk) 15:08, 19 January 2018 (UTC)

-------------------------------------- edits

What's up with those super long dashed lines? Doesn't seem standard? Nikolaih^☎️📖 23:34, 14 January 2021 (UTC)

Nikolaih, I have removed these dashed lines, but much more edits are needed to be conform to the manual of style. D.Lazard (talk) 10:31, 15 January 2021 (UTC)

Rewrite proof!

The proof of Cauchy-Schwarz is one of my favs; how unpleasant to stumble upon such unnecessarily lengthy and ugly proofs. I humbly suggest that all proofs be removed from the article, to only keep one: the function $p(t)=\langle tu v,tu v\rangle =\Vert u\Vert ^{2}t^{2} 2t\langle u,v\rangle \Vert v\Vert ^{2}$ is a second degree polynomial, moreover it is always nonnegative, so its (reduced) discriminant $\Delta /4=\langle u,v\rangle ^{2}-\Vert u\Vert ^{2}\Vert v\Vert ^{2}$ must be nonpositive. Seub (talk) 13:53, 7 September 2021 (UTC)

Go right ahead! I support your argument. Proof must be simple. And nobody will learn from this ugliness anyway. I would fix this myself, but I don't have time right now. StrokeOfMidnight (talk) 14:20, 7 September 2021 (UTC)

By the way, your proof doesn't work over $\mathbb {C} .$ StrokeOfMidnight (talk) 19:55, 7 September 2021 (UTC)

R² proof vs Rⁿ proof

The proof given for $\mathbb {R} ^{2}$

If $\mathbf {v} =\left(v_{1},v_{2}\right)$ and $\mathbf {u} =\left(u_{1},u_{2}\right)$ then the Cauchy–Schwarz inequality becomes: $$\langle \mathbf {u} ,\mathbf {v} \rangle ^{2}=(\|\mathbf {u} \|\|\mathbf {v} \|\cos \theta )^{2}\leq \|\mathbf {u} \|^{2}\|\mathbf {v} \|^{2},$$ where $\theta$ is the angle between $u$ and $v.$

Does this proof not work for $\mathbb {R} ^{n}$ as well? — Preceding unsigned comment added by Gabiteodoru (talk • contribs) 10:58, 13 February 2022 (UTC)

Cosines cannot be used for proving the inequality since the angle of two vectors is defined by $\cos \theta ={\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\sqrt {\|\mathbf {u} \|\|\mathbf {v} \|}}},$ and this is the inequality that asserts that the absolute value of the cosine is at most 1. So, using cosines leads to a circular reasoning. "The inequality becomes" means that this is not supposed to be a proof. D.Lazard (talk) 11:56, 13 February 2022 (UTC)

Thanks a lot for explaining, I actually never realized that this inequality was tied to the output range of the cosine function. Then I shall reformulate my question -- the expression "the inequality becomes" does hold for $\mathbb {R} ^{n}$ as well, does it not? I'm puzzled why it appears in a $\mathbb {R} ^{2}$ specific section -- is there anything else I might be missing? Thanks! Gabi Teodoru (talk) 23:28, 13 February 2022 (UTC)

You are perfectly correct. More, the formula written as $\cos \theta =\cdots$ is the standard way for computing angles in space geometry. As far as I understand, this section is here for motivating beginners who know only plane geometry. It is clear that this section requires clarification and other improvements. D.Lazard (talk) 08:49, 14 February 2022 (UTC)

Proof 3

I have found a correct proof for the general Cauchy-Schwarz inequality. I attempted to add it as "Proof 3" under proofs, but another user insists that this is not allowed. While original research is generally not allowed on wikipedia, the results of routine mathematical calculations are allowed, provided that there is editor consensus that the calculations are correct. The person who is removing my additions seems to agree that the calculation is correct, so I see no need to remove the proof. Chipdink69420 (talk) 16:51, 9 August 2023 (UTC)

Your proof is probably correct but it is not sufficient for including it. WP:CALC says that routine calculations need not to be sourced. This applies to each step of your proof, but not to the whole proof. So, a WP:reliable source is required for allowing readers to be sure it is correct, especially when they have not the competence to verify it by themselve. Also, there are already two proofs, and it is to you to convince the other editors that it is worth to include a third one. IMO, your proof is confusing for many readers, because of the number of auxiliary variables that are used. D.Lazard (talk) 17:38, 9 August 2023 (UTC)

Four proofs

Do we need to have four separate proofs? Currently we have:

• "Proof 1", showing that C-S follows from a rather complicated-looking equality. A benefit is that one also proves this equality, but there's not any intuition given for what this equality means and I don't know that it's useful in its own right.
• "Proof 2", with the idea of projecting and using the Pythagorean theorem. The computation is somewhat less complicated than Proof 1 and also gives some geometric intuition.
• "Proof for real inner products", an easy argument from computing the discriminant of a non-negative quadratic. (Terry Tao describes this technique as "arbitraging" asymmetries in the trivial inequality $\langle v w,v w\rangle \geq 0$.) Despite the title, this proof can work just as well for complex inner products; I added a note about this to the article but didn't change the section title. This is the standard proof found in textbooks like Rudin.
• "Proof for the dot product". I think this is essentially the same quadratic-discriminant proof as the previous, but it's somewhat obfuscated by writing out a bunch of terms of the sums. This is the least applicable of the proofs, but has the benefit that it could be understood without knowing the definition of an inner product.

I think we should keep "Proof 2" and "Proof for real inner products", but remove "Proof 1" as it's overcomplicated and doesn't provide much intuition. I would also like to remove "Proof for the dot product" as well, though there is some cost to doing so. I think we should also rename "Proof for real inner products" but I'm not sure to what. Elestrophe (talk) 16:44, 15 February 2024 (UTC)

I've gone ahead and done this for now. Elestrophe (talk) 20:44, 29 February 2024 (UTC)