Talk:Euclidean space/Archive 2

triangular inequality should be triangle inequality
"Nobody" says or writes "triangular inequality" anymore. Everybody says "triangle inequality". Even "triangular inequality" redirects to "triangle inequality". If a student looks this up, he or she should be introduced to modern terminology. LMSchmitt 07:23, 1 November 2020 (UTC)
 * This seems simply a bad translation from French. I have fixed it. In any case, as redirects to Triangle inequality, you could have made the change yourself. D.Lazard (talk) 08:45, 1 November 2020 (UTC)

Def of linear function
To verify that a function is linera, it is enough to show, and more effective and simpler to show

f(a x + y)=a f(x) + f(y)

this with y=0 and a=1 implies preservation of vector space ops. The two scalars mu and nu that are used in the appended proof of linearity for an isometry are therefore just an incarnation of endless copy/paste of patterns from 1800 through high school through many books. The second scalar in proofs of linearity (here mu) can always be dropped (to 1), and saves writing efforts, energy and time. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 07:09, 1 November 2020 (UTC)
 * , It is true that your preferred definition of linearity is sometimes simpler to verify. However, it destroys the symmetry and the homogeneity of the formula, and this is not always a good idea. In particular, if you try to expand the proof sketched in the footnote, the homogeneity may be helpful for checking the computation, and, possibly, finding where an error has been done (all terms of the expanded formulas must have the same degree in the scalars). D.Lazard (talk) 09:06, 1 November 2020 (UTC)


 * . As I tried to indicate before, in my opinion, your favorite method comes from school indoctrination and was passed along for many generations.
 * [A] The proposed method ("my") is always simpler to verify. 40 years teaching math at university level tell me that. In math, simplicity wins, and "my" method is simpler, because it uses fewer symbols.
 * [B] What is the symmetry/homogeneity that you endorse good for.? If you need the homogeneity for that little proof, then you can't do elemental algebra. The most nasty proof which involves ax+y is in proofs of multi-linearity for tensor product constructions in linear algebra, which one does effectively by showing linearity in every "component" (every tensor factor). In that case, ax+y is much better to write on the blackboard than ax+by. ax+by just makes the computation more messy and more complicated to write and prone to writing errors. I would appreciate one proof in one book, video, anything where the symmetry really is of good use. I haven't seen one ever. I find what you put forward as your valid opinion is actually void from my experience. Please, observe that convexity uses (1-a) and a, but that's an application, and f(ax+by) = a f(x) + b f(y) follows from the simpler formula. This is about verifying linearity.
 * [C] If you are dealing with something really complicated, prove the formulas for a* and + separately like one verifies the properties of a norm.
 * LMSchmitt 11:02, 9 November 2020 (UTC)
 * Our respective backgrounds have nothing to do here (for your information, I am emeritus professor in mathematics and computer science). The best way of defining linearity in general is an interesting question, which is out of scope here. Here we have a proof that is sketched in a footnote. Such a proof is not fundamental as it could be replaced by a reference to a textbook. So it has to be sketched for providing only the main idea, leaving the details to the readers. Using your above non-homogeneous formula does not really simplifies the computation needed for verifying the details, and would make more difficult to fix possible errors during this computation. In any case, as the formulation is correct, your personal preferences is not a good reason for changing it (see MOS:VAR, for example). D.Lazard (talk) 11:45, 9 November 2020 (UTC)

The way to decisively win this argument is to marshal reliable sources and show that they overwhelmingly favor one formulation or the other. And if that's how you want to contribute to Wikipedia, then that's your right. But please consider whether there are more important tasks to spend our time on, than a disputed micro-improvement to a footnote that might be deleted six months from now anyway. Mgnbar (talk) 13:11, 9 November 2020 (UTC)

A bad sentence
This reduction of geometry to algebra was a major change of point of view, as, until then, the real numbers—that is, rational numbers and non-rational numbers together–were defined in terms of geometry, as lengths and distance.

This sentence is bad. There are other non-rational numbers (complex numbers) which are not real numbers. In addition, lengths and distances are always positive. Real numbers such as -1 are not. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 13:27, 25 October 2021 (UTC)
 * This is true that there are negative numbers and complex numbers. But at Descartes' time, complex numbers did not really exist (this is why Descartes call them "imaginary"). About negative numbers, they were (and still are) defined from positive numbers. So, it is true that before Descartes, non-integer numbers were defined from geometry. D.Lazard (talk) 16:04, 25 October 2021 (UTC)


 * "the real numbers were defined as lengths and distance" is the core sentence above. And it is WRONG (see: -1). In the present sentence, there are no 'mentioned exceptions' or 'significant qualifiers'. It also should be "distances".


 * The statement "About negative numbers, they were (and still are) defined from positive numbers." is just BULL for the sake of being "right" only. And that doesnt mean they are all positive.


 * The real numbers (IR,+) can be defined as the Grothendieck group of the positive real numbers. Nobody does that. One common approach is to see them as the abstract completion of the rational numbers (equivalence classes of Cauchy sequences of rational numbers) which immediatly gives + and *. Another approach is to postulate an ordered field with the SUP property. Both these standard approaches start from a field which per definition already contains negative numbers. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 21:59, 25 October 2021 (UTC)

"Euclidean norm" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Euclidean norm and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 May 4 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. fgnievinski (talk) 16:58, 4 May 2022 (UTC)