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The assessment of Lagrange's identity and Pythagoras' theorem

David: Although I have vacillated on this subject, I believe the logic here runs like this:

• Multiplication table → Lagrange's identity in the simplified form. Without some way to connect the vector components of a and b to a × b, one cannot connect the special form of Lagrange's identity to the cross product.
• Such a multiplication table doesn't exist except in 3D and 7D.
• It is invariable true that sin² +cos² = 1, which follows from the definition of these functions, and is quite independent from their origin in the plane.
• Lagrange's identity takes on the special form in 7-D and 3-D because a suitable multiplication table exists.
• Substitution of the dot product in terms of cosine then leads inevitable to the magnitude of the cross product as "ab sin θ"

Based upon the above observations, the fact that the Jacobi identity does not apply in 7D has no bearing upon ||a × b|| = ab sin θ . Brews ohare (talk) 18:04, 25 May 2010 (UTC)

Brews, Let's go right back to Brougham Bridge in 1843. Sir William Rowan Hamilton supplied us with the embryo of what Josiah Willard Gibbs eventually presented as,

c = a × b	a × b
i	j×k
j	k×i
k	i×j

If we multiply out vectors a and b using this table, we end up with,

$|\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\$

If we already assume 3D geometry and the cosine relationship in the dot product, then we can use,

sin² +cos² = 1

to obtain the sine relationship for the cross product. In most applied maths courses, we are only concerned with the 3D cross product, and as such, the cross product is usually defined straight away in terms of the sine relationship, and in terms of its connection with the area of a parallelogram. But as you know, it is possible to use,

$|\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\$

as a starting point. Silagadze does this and shows that apart from the trivial cases of 0, and 1 dimensions, that a cross product will only exist in 3 or 7 dimensions. A seven-dimensional cross product cannot automatically be assumed to link to the sine of an angle. Indeed, the Jacobi identity, which onlys applies in 3D, can be shown to be linked to the sine of the angle. This has been done on your talk page, and it tends to suggest that the sine relationship is restricted to 3D.

If we can only link the 3D case to the sine of an angle, then Pythagoras's theorem is shown to be the Lagrange identity only in three dimensions. David Tombe (talk) 19:44, 25 May 2010 (UTC)

Brews, Having now looked at your recent edits at the 'Lagrange's identity and Pythagoras's theorem' section, I would raise two separate issues. The first is that the emphasis of that section has now been changed to 'cross product' and its magnitude, whereas before, the cross product was only an intermediatory mechanism used to demonstrate that Pythagoras's theorem is the special 3D case of the Lagrange identity.

The second issue is of course over the 7D question. Your logic from the first line assumes that everything holds equally for the 3D case and the 7D case. Because of the Jacobi identity and its restriction to 3D, the 7D case must be dealt with separately. David Tombe (talk) 20:03, 25 May 2010 (UTC)

Hi David: Yes, the emphasis has changed.

The logic is not quite as you say. The logic is that for some choices of multiplication rule for e_i × e_j , Lagrange's identity takes on a special form. I rely upon Lounesto as the source saying that such a multiplication table cannot be found except in 3D and in 7D. Would you agree that this form does result for the stated multiplication tables in 3 D and in 7D?

Once you have established the multiplication table, got the special form, then it results that a × b has magnitude ab sin θ. Would you agree with that?

Thus, the limitation on dimensions is traced back to the ability to find the appropriate multiplication table to achieve the special form of Lagrange's theorem. I've not tried to track down its origins. But ultimately I am relying upon Lounesto, and hope I have read him correctly. Because a multiplication table can be found for 7-D where the Jacobi identity doesn't work, I'd say that the Jacobi identity is not the critical factor.

Can you explain whether you have issues with any of this? Brews ohare (talk) 23:15, 25 May 2010 (UTC)

Incidentally, this google spreadsheet evaluates Lagrange's theorem as a sum of product differences and as the magnitude of x × y and also checks that x × y is orthogonal to x and to y in seven dimensions. Thus, IMO it is established for every numerical example you want to plug into this spreadsheet that the "Pythagorean" form of Lagrange's identity holds true for the chosen multiplication rule. However there are several such multiplication tables that work, not just Lounesto's, I'd guess, because my choice, Blackburne's choice and your own choice are different and all work. Brews ohare (talk) 23:10, 25 May 2010 (UTC)

Brews, Let's consider your statement,

The logic is that for some choices of multiplication rule for e_i × e_j , Lagrange's identity takes on a special form.

In my view, the emphasis should not be on the definition or magnitude of the cross product. In fact, I don't think that the magnitude of a cross product needs the Lagrange identity. It's more a case of that Lagrange's identity takes on the particular form,

$|\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\$

in the special cases of 3 and 7 dimensions. It's trivial to prove this in the 3D case. In the 7D case the proof is much more extensive and involves the mutual cancelling of 168 out of 252 terms, and the anti-distributing of the remaining 84 terms into 21 squared terms in brackets. The concept of angle needn't come into it at this stage. Angle comes later.

But if we assume that the cosine and sine relationships hold in the dot product and the cross product respectively, then Pythagoras's theorem follows. As such, Lounesto would be absolutely correct in referring to this form of the Lagrange identity as the Pythagorean identity. But the Jacobi identity probably rules out the sine relationship in the 7D case. Hence I would say that Lounesto may only be half-right in calling this form of the Lagrange identity the Pythagorean identity. David Tombe (talk) 23:44, 25 May 2010 (UTC)

David:

You object that the primary result is that Lagrange's identity takes a certain form. But it cannot take on any relation to the cross product until a connection is made between the cross product and the components of a and b. That connection is a cross-product definition, and also defines a multiplication table for the unit vectors. The Lagrange identity then can be related to the choice of cross product, whatever way it has been chosen. Angle is not engaged in forming the particular multiplication table for e_i × e_j in either 3-D or 7-D.

Only in 7D and 3 D can a connection between a and b be found such that Lagrange's identity in terms of the new a × b takes the form: $|\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\$. No angle involved here, only a relation between components of a and b and a×b. In 7D and 3D we can find some multiplication table that converts Lagrange's identity to $|\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\$. Combination of that result with a.b = ab cos θ leads to the result ||a×b|| = ab sin θ. The angle is introduced through the dot product, and brought to apply to the cross product through Pythagoras' trigonometric identity. If there is a clear flaw in this argument, please underline it for me. Brews ohare (talk) 04:15, 26 May 2010 (UTC)

BTW, the multiplication for cross product also is subject to the restriction that a × b is orthogonal to both a and b. Brews ohare (talk) 04:15, 26 May 2010 (UTC)

Brews, I agree with all of those points above, except for the issue about the angle. But for the purposes of this article, which is about 'Pythagoras's theorem', the only significant fact is that,

$|\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\$,

holds only in 3 and 7 dimensions. The justification of this fact is something that really only needs to be treated in detail in the two articles about 'cross product', and not in this article. Lounesto calls this equation the 'Pythagorean identity'. If we assume the cosine relationship for the dot product, then the sine relationship for the cross product falls out automatically, just as you have said above. And Lounesto does assume it. But in the Lounesto reference which you supplied above (Page 96 section 7.4) [1], Lounesto also points out in the same paragraph that the Jacobi identity doesn't hold in the 7D case. That puts a question mark over both the sine and cosine relationships in the 7D case, since the Jacobi identity can be shown to be linked to the sine relationship in the cross product. David Tombe (talk) 10:50, 26 May 2010 (UTC)

Proofs

What do you think about including this proof? It is very short and intuitive. Better than any of the algebraic ones in the article, IMHO. Tercer (talk) 04:55, 28 May 2010 (UTC)

Very cute indeed. If you find it in an article or in a text book, It would be a good addition, and I'd put it first. I have had a look at some of the comments in the blog, trying to find a proper source. They mention Polya's "Mathematics and Plausible Reasoning" and Barenblatt's "Dimensional analysis". I checked them, but it turns out out that they don't have this proof. They merely mention the similarity of the 3 triangles. DVdm (talk) 08:11, 28 May 2010 (UTC)

Two proofs are provided at the link given. The first is another instance an application of this one in the article already. The second involves knowing that det ( O ) = det ( R O R^-1), which is cute, but requires a basis be laid. For example, the rotation matrix R:

$R={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}\ ,$

has determinant 1, and det (ABC) = det(A) det(B) det(C).Lancaster This approach to a proof uses the Pythagorean trigonometric identity, which makes it circular, I'd say. Brews ohare (talk) 14:31, 28 May 2010 (UTC)

The first is indeed a special case of the one you linked; but this is a strong point. It obviates the need of a more sober source (although I'd trust Terry Tao's blog before any article), and you can reference the special case when proving the generalisation, from within the article. Although... the general proof presented here is not well-written. It makes the notation heavy by unnecessarily specifying the proportionality constant. And there's a typo.

The second is indeed not a good one. It can be rewritten to avoid circularity, but that would make it very long and defeat the purpose. Tercer (talk) 16:27, 28 May 2010 (UTC)

This proof using similar triangles is a special case of the proof in the article that Brews referred to in his message above. It's just putting the similar triangles inside the large triangle instead of outside. --Bob K31416 (talk) 16:38, 28 May 2010 (UTC)

The starting triangle can be formed by folding triangle A along side a and triangle B along side b; the starting triangle also is identical to triangle C, which just flips the starting triangle along side c. By the arguments given for similar figures in this one the areas are proportional to the sides along the interior triangle. Hence, for some proportionality constant α, we have α (a² + b²) = αc², and α cancels out. It might be worth adding this example in this same section, no? Brews ohare (talk) 17:00, 28 May 2010 (UTC)

No. : ) --Bob K31416 (talk) 17:06, 28 May 2010 (UTC)

BTW if you reoriented triangles A and B, you could flip them about their hypotenuses so that they fit into C (start). But still no. : ) --Bob K31416 (talk) 17:12, 28 May 2010 (UTC)

oooo, I see you almost did it, but not quite. You have to reorient A. Still no. : ) --Bob K31416 (talk) 17:15, 28 May 2010 (UTC)

Got it. Brews ohare (talk) 17:23, 28 May 2010 (UTC)

Congratulations! I have to hand it to you, you sure work fast with those images. But I don't think it's worth adding to the section. However, your image does demonstrate that the proof discussed in this talk section is a special case of the proof that is already in the article. Regards, --Bob K31416 (talk) 17:28, 28 May 2010 (UTC)

BTW, Tercer, can you explain to me how the matrices:

${\begin{bmatrix}a&b\\-b&a\end{bmatrix}}\ ,$

and

${\begin{bmatrix}c&0\\0&c\end{bmatrix}}\ ,$

are mapped onto these triangles in the second proof? Brews ohare (talk) 17:00, 28 May 2010 (UTC)

I'm not sure what you mean by mapping onto the triangles. The idea of the proof is for you to notice that the matrix ${\begin{pmatrix}a&b\\-b&a\end{pmatrix}}\ ,$ is only a rotation composed with enlargement by c, i.e., ${\begin{pmatrix}c&0\\0&c\end{pmatrix}}{\begin{pmatrix}a/c&b/c\\-b/c&a/c\end{pmatrix}}$. But this is only true when a and b are the corners of a right triangle, so that a/c and b/c are actually a cosine and a sine. Is this what you asked?

If you want something a little more geometric, apply both transformations to the unit square. The c transformation will give you a c by c square, of area c², and the other will give you this square rotated. It is easy to see that this square is composed by four right triangles of corners a and b plus a square of side (b − a), so that its area will be $4{\frac {ab}{2}}+(b-a)^{2}=a^{2}+b^{2}$ If you agree that rotations shouldn't change area, the claim follows. Tercer (talk) 03:35, 2 June 2010 (UTC)

Tercer: I think you understand my problem, how to connect the two matrices to some geometric statement, but I don't understand the answer. To be specific, if my right triangle has vertices at (0,0) (0,a) (b,0) what rules of correspondence lead me to identify this triangle with the matrix ${\begin{bmatrix}a&b\\-b&a\end{bmatrix}}\ ,$ and next, how do these same rules lead to the triangle identified with matrix ${\begin{bmatrix}c&0\\0&c\end{bmatrix}}\,$ as the same triangle, but rotated? See my confusion? Brews ohare (talk) 11:20, 2 June 2010 (UTC)

Orientation of a square of side c described by inserted right triangles of sides a and b.

Perhaps this figure can assist. We adopt the convention of describing a square by a matrix consisting of the coordinates of its lowermost and uppermost corners. For the top square, the description is ${\begin{pmatrix}b&-a\\a&b\end{pmatrix}}\ ,$ As the square becomes vertically aligned, side a of the triangle describing its orientation goes to zero, and side b goes to c. Hence the description of the bottom square is ${\begin{pmatrix}c&0\\0&c\end{pmatrix}}\ .$ We now have the required matrices mapped to the geometry. Apparently the area of the bottom square is the determinant of its matrix. Assuming that holds for the top square too, we get its area as b^2 + a^2. Because rotation cannot change area, we have c^2 = a^2 + b^2. Whaddaya think? Brews ohare (talk) 17:32, 2 June 2010 (UTC) The unresolved element is: How do we go about proving the determinant is the area, regardless of orientation? One must be careful not to beg the question. It can be done algebraically, as the area of the top figure is the sum of four triangles (2ab) and the center square (b-a)^2 so c^2 = b^2+a^2 regardless of orientation. But that doesn't use properties of matrices and determinants to get Pythagoras, but an algebraic proof of Pythagoras to prove the properties of the determinant. Brews ohare (talk) 17:47, 2 June 2010 (UTC)

Figure for "Law of cosines"

The separation s of two points (r₁, θ₁) and (r₂, θ₂) in polar coordinates is given by the law of cosines.

The angle in this figure is the difference between θ₁ and θ₂. Although θ is measured counterclockwise, this difference is simply a number and has no polarity. Brews ohare (talk) 21:01, 1 June 2010 (UTC)

I notice that you made the arrow point in both directions now. The figure explicitly mentions polar coordinates where indeed θ₁ and θ₂ are measured counterclockwise. In this context angles have an orientation, so the difference between two of them, defining an angle as well, does have an orientation. It is of course of no importance for the equations in the text, but in the figure it looks important to have it correct.

Other opinions? DVdm (talk) 21:18, 1 June 2010 (UTC)

DVdm: Thanks for suggesting a change, although perhaps I didn't do quite what you had in mind. My thought is that the double arrowhead indicates a dimension, which is correct for this diagram, where the focus is upon the interior angle. The alternative would seem to require θ₁ and θ₂ be indicated with individual arrowheads in the counterclockwise direction, which makes for added lines and clutter and distraction, IMO. Brews ohare (talk) 21:33, 1 June 2010 (UTC) I modified the diagram to identify the difference Δθ = θ₁ − θ₂. Brews ohare (talk) 22:01, 1 June 2010 (UTC)

Distance in various coordinate systems

The subject of this section seems off topic to me.

There is a Distance article where this subject would be more on topic.

The former version where it was briefly noted that the distance formula is an application of the Pythagorean theorem seemed fitting. Trying to make the theorem fit in other coordinate systems seems wide of the mark.

Calling the law of cosines "the Pythagorean theorem in polar coordinates" seems incorrect to me.

The special case where the difference in angular coordinates is 90° is simply a right triangle. That polar coordinates are used seems irrelevant.

Reducing the law of cosines for the case where the angle is 90° would not seem to produce "the usual form of Pythagoras' theorem" because I don't consider the law of cosines to be a "form of Pythagoras' theorem." -Ac44ck (talk) 05:26, 2 June 2010 (UTC)

Hi Ac44ck: The topic "distance in various coordinate systems" is possibly the most common use of Pythagoras' theorem in everyday activities like carpentry or surveying. The extension to n dimensions is more abstract, but not a lengthy digression. It is a rather important generalization in mathematical applications.

The general methodology for any curvilinear coordinate system in any number of dimensions is mentioned: namely, just use the definitions of the coordinates in terms of the Cartesians and plug them into the "standard" Cartesian form for the separation. The resulting odd forms look very different from Pythagoras' theorem, but the reader may be reassured that it still underlies them.

I judge it to be helpful to the reader to supplement words with an example, and the polar coordinate case is about as simple an example as can be found. Polar coordinates are interesting in themselves. They show the Pythagorean distance as a sum of squares in Cartesian coordinates contains within it the seed of the generalization to arbitrary triangles, namely the law of cosines.

I've modified the text to be clear that it doesn't say that "the law of cosines ‘is a form of Pythagoras' theorem.’" The text also says that the law of cosines "generalizes" Pythagoras' theorem. I hope that now the wording is beyond ambiguity.

The changes in text suggested by your comment improves matters, thank you, and I hope you see some purpose in this section now. Brews ohare (talk) 12:22, 2 June 2010 (UTC)

It is an interesting topic, and the image is well done. And the mention of distances in Cartesian coordinates (in various dimensions) seems fitting.

The small number of hits here suggests to me that the term as applied in the article is not widely used:

http://www.google.com/search?q=%22Generalised+Pythagorean+Theorem%22+cosines

I would be surprised if the Pythagorean theorem underlies distance formulas in general. The great-circle distance article says:

Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a different form.

The Pythagorean theorem may be analogous to other distance formulas, especially that for polar coordinates. But to say it "underlies" distance formulas in general may be a stretch.

To my eye, the distance formulas in the great-circle distance article bear little resemblance to the Pythagorean theorem.

One might calculate the distance "as the crow flies" (or maybe better, "as the mole burrows") between two points on the earth by translating between Cartesian and spherical coordinates. But in situations where it is convenient to be working in spherical coordinates, I would think one would have rare occasion to seek a literal straight-line distance between two arbitrary points. For more exotic geometries, literal straight-line distances may be needed even less frequently. -Ac44ck (talk) 06:37, 3 June 2010 (UTC)

Don't forget that in non-Euclidean geometries distances are a result of integrated infinitesimal straight-line distances, which are expressed by the standard or by the generalized Pythagoran theorem, applied to infinitesimal coordinate differences. DVdm (talk) 09:24, 3 June 2010 (UTC)

"Law of cosines", the term preferred to "Generalized Pythagorean theorem", gets 94,800 hits. The problem of finding the side of triangle that is not a right triangle comes up frequently. The straight-line distance between two points is fundamental in applications, for example see this. "Straight-line distance" comes up with almost 4 million hits. The topic of great circles etc. comes under "Non-Euclidean geometry", another subsection. Non-Euclidean geometry is not the same subject as choosing various coordinate systems in Euclidean space. The coordinate system is chosen to suit the problem, but it doesn't change the space from Euclidean. So the orbit of a planet might be calculated in elliptic coordinates, but it moves under a straight-line force law. Brews ohare (talk) 15:12, 3 June 2010 (UTC)

My point in reporting Google hits was that few seem to use the term "Generalized Pythagorean theorem" for anything. It doesn't suggest that many consider the term to be a synonym for the law of cosines. No doubt the topic of straight-line distance is important. That's why there is a distance article. This is not that article. As for orbital coordinates, the radial distance between the sun and planet is an input, not an output. -Ac44ck (talk) 07:53, 4 June 2010 (UTC)

Ac44ck: Your statement: “As for orbital coordinates, the radial distance between the sun and planet is an input, not an output.” confuses me. The point isn't about input and output; it is about changing coordinate systems from the Cartesian to curvilinear, and expressing the straight-line distance in curvilinear coordinates. This link is one example. I think you are suggesting that somehow straight-line distance in curvilinear coordinates is of no consequence, but I am unsure how you support your point.

I was impressed that the Pythagorean theorem applied to distance in Cartesian form (which is the right-triangle theorem) produced the generalization of this form of Pythagoras' theorem to the law of cosines when polar coordinates were substituted for the Cartesian. That seems to me an interesting result of this example, quite apart from its utility as a concrete illustration of how Pythagoras' theorem can be used in general curvilinear coordinates.

IMO the sentence about how Pythagoras' theorem is employed in curvilinear coordinates is useful, and the polar coordinate example serves a double purpose in demonstrating the method and in deriving the law of cosines. Brews ohare (talk) 14:46, 4 June 2010 (UTC)

I don't think there are "forms" of the Pythagorean theorem. There is only one Pythagorean theorem. That's why it has a name. The law of cosines resembles it.

This article is about the Pythagorean theorem. Some have argued that the theorem is more about areas than distances. I don't see that a discussion "about changing coordinate systems from the Cartesian to curvilinear, and expressing the straight-line distance in curvilinear coordinates" is relevant here.

Deriving the law of cosines is interesting ... and relevant in the article about the law of cosines.

It is possible to build a complex formula for the straight line distance across a sphere using values for latitude, longitude and radius. But is it practical? Wouldn't it be easier to convert the spherical coordinates to Cartesian coordinates and apply the simpler distance formula to rectangular coordinates? How is it useful to mention general curvilinear coordinates in this article about the Pythagorean theorem? -Ac44ck (talk) 19:17, 5 June 2010 (UTC)

Ac44ck: It is useful because Pythagoras' theorem directly determines distance in Cartesian coordinates, and one might reasonably anticipate the question of how it translates into other coordinate systems. Likewise, Pythagoras' theorem applies to right triangles, and one might anticipate the question of how it can be altered to work for triangles that are not right. Likewise, Pythagoras' theorem arose in two dimensions, and one might anticipate the question of how it can be adjusted to apply in 3 dimensions, or in n dimensions. These queries and others need not be exhaustively handled, and a subsection need only provide some flavor of the answers and some links where more extensive discussion may be found. That is what has been done in the various sub-sections. If you personally have no curiosity about these questions, that doesn't mean there are no such readers; only that you are not among them. Brews ohare (talk) 19:51, 5 June 2010 (UTC) IMO a major utility of WP is to assist a reader making an inquiry to broaden their perspective and to see connections they may not have thought about. WP is more like a Thesaurus than a dictionary. It doesn't have to pare things down to the barest discussion possible, but to the contrary, should be an eye-opener to related matters. That is why people use WP despite doubts as to its accuracy: it is a great starting point to shake down a subject. Brews ohare (talk) 19:59, 5 June 2010 (UTC)

Lagrange's identity

I drastically reduced this section and deleted Lagrange's identity from its header. I believe its previous form was misleading and too long.

The role of Lagrange's identity is not to establish the result:

$\|\mathbf {a} \times \mathbf {b} \|^{2}+(\mathbf {a} \cdot \mathbf {b} )^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2},$

which is better viewed as part of the definition of a cross product. For example see: WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10): 697–701. {{cite journal}}: Unknown parameter |month= ignored (help)

Rather, it's role is to establish a further limitation upon the cross product, namely:

$\|\mathbf {a} \times \mathbf {b} \|^{2}=\sum _{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2},$

where n = 3 or 7. It is this identity that restricts the multiplication tables for the unit vectors $\mathbf {\hat {e}} _{i}\mathbf {\times {\hat {e}}} _{j}$ determining the cross product:

$\mathbf {a\times b} =\sum _{i}a_{i}\mathbf {\hat {e}} _{i}\mathbf {\times } \sum _{j}b_{j}\mathbf {\hat {e}} _{j}=\sum _{i}\sum _{j}\ a_{i}b_{j}\ \mathbf {\hat {e}} _{i}\mathbf {\times {\hat {e}}} _{j}\ .$

The cross products of the unit vectors ${\mathbf {\hat {e}} _{i}}$ are forced by Lagrange's identity to satisfy very specific multiplication tables. In 3-dimensions this table is:

$\mathbf {\hat {e}} _{i}\mathbf {\times {\hat {e}}} _{j}=\mathbf {\hat {e}} _{k}\ ,$

with {i, j, k} a cyclic permutation of {1, 2, 3}. In 7-dimensions, the multiplication is provided by [See

Door Pertti Lounesto (2001). Clifford algebras and spinors (2nd ed.). Cambridge University Press. ISBN 0521005515.]:

$\mathbf {\hat {e}} _{i}\mathbf {\times {\hat {e}}} _{i+1}=\mathbf {\hat {e}} _{i+3}\ ,$

with {1, 2, 3, 4, 5, 6, 7} permuted cyclically and translated modulo 7. Brews ohare (talk) 17:50, 4 June 2010 (UTC)

Second thoughts

I'm going to have to revise my thoughts above: the Lagrange identity means that:

$\|\mathbf {a} \times \mathbf {b} \|^{2}+(\mathbf {a} \cdot \mathbf {b} )^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2},$

and

$\|\mathbf {a} \times \mathbf {b} \|^{2}=\sum _{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2},$

are in fact the same restriction upon the cross product, and either may be used equally [in 3D and 7D. --added note. Brews ohare (talk) 13:31, 5 June 2010 (UTC)]. It is this restriction in either of its forms, plus the restrictions requiring orthogonality of the cross product to its constituents, that lead to the multiplication table requirements and the dimensional restrictions to n=3 or 7. Brews ohare (talk) 00:41, 5 June 2010 (UTC)

Brews, Another way to look at it is that Lagrange's identity and vector cross product only intersect at 3 and 7 dimensions. Don't forget that we can still actually have a 5D cross product,

z = x × y	x × y
i	j×m, l×k
j	i×l, k×m
k	i×j, l×m
l	i×m, j×k
m	i×k, j×l

but that it won't be compatible with the Lagrange identity. The "miraculous cancellation" only occurs in the special case of 7D. It follows therefore that the distributive law only holds in cross products of 3 and 7 dimensions, and hence a 5D cross product would be useless because we would not be able to multiply it out into its components.

Nevertheless, we can have 'cross product' in 5D, we can have the Lagrange identity in 5D, and there are some who believe that we can have Pythagoras's theorem in 5D. But one thing is certain, and that is that when we introduce the Jacobi identity, all these concepts form a single package in 3D with no specific starting point for the inter-relationships. Outside of 3D, the package falls apart. David Tombe (talk) 12:21, 5 June 2010 (UTC)

Thanks, David. I wasn't too careful in my remarks above. I meant to say the two formulations were the same for 3-D and 7-D, not for general dimensions. Brews ohare (talk) 13:29, 5 June 2010 (UTC)

Brews, I think that alot of the problem with all of this was finding the starting point. It seemed that no matter where one started, that the proof was merely a tautology. I eventually stood back from it all and I could see a self contained package of 3D geometry, Pythagoras's theorem, rotations and angle, Lagrange identity, cross product, dot product, and Jacobi identity. The special case of 7D will always remain somewhat of a mystery, because the 'miraculous cancellation' allows the cross product to fit with the Lagrange identity. I don't yet know what the full significance of this is. David Tombe (talk) 15:55, 5 June 2010 (UTC)

Non-Euclidean geometry

What I perceive of as off topic seems to go back quite a while. A version from July of 2009 contains this section:

The Pythagorean theorem in non-Euclidean geometry

It starts out:

The Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry.

The notion of a "form" of the Pythagorean theorem is there. There are "analogies to the Pythagorean theorem," but I don't think of them as "forms of." After saying it "does not hold in non-Euclidean geometry," it then states,

For any right triangle on a sphere of radius R, the Pythagorean theorem takes the form ...

Not an analogy of the law of cosines, but a form of the much more specialized Pythagorean theorem. It then says,

This equation can be derived as a special case of the spherical law of cosines.

All very interesting, but wide of the mark. The Pythagorean theorem is a special case of the law of cosines. The spherical law of cosines is analogous to the planar law of cosines. To it all turn it on its head and say that the spherical law of cosines is a "form of the Pythagorean theorem" seems daft. This is why people distrust Wikipedia: it gets things serially backwards, they can stay that way for a year, and more-of-the-same gets added because it is fueled by precedent. -Ac44ck (talk) 07:12, 8 June 2010 (UTC)

@Ac44ck: Please allow me to jump in here for a second? A bit higher up, you wrote:

The small number of hits here suggests to me that the term as applied in the article is not widely used:

http://www.google.com/search?q=%22Generalised+Pythagorean+Theorem%22+cosines

This gave 6 results, which is a small number indeed. However, have you tried it this way?

http://www.google.com/search?q=%22Generalized+Pythagorean+Theorem%22+cosines

and this way, with google books?

http://books.google.com/books?&q=%22Generalized+Pythagorean+Theorem%22+cosines

DVdm (talk) 07:56, 8 June 2010 (UTC)

Ac44ck: The topic has now shifted to a different subsection. The theme of your remarks has some similarities, though. Perhaps you would be happy if the wording "Pythagorean form" was replaced? I'll do that. Let me know your reaction, please. Brews ohare (talk) 14:20, 8 June 2010 (UTC)

How bad was it before?

This article is nearly nine years old. Has it been so broken that it needs this flurry of activity to fix it?

http://toolserver.org/~daniel/WikiSense/Contributors.php?wikilang=en&wikifam=.wikipedia.org&grouped=on&page=Pythagorean_theorem

Edits           User            first edit              last edit
405 (244/156)   Brews ohare     2010-04-24 15:25        2010-05-22 04:01
216 (138/78)    Michael Hardy   2003-03-26 02:30        2010-05-09 03:00

The greatest number of revisions by a single editor have been made in less than a month. The editor with the second highest number of edits took more than seven years to accumulate half that number. - Ac44ck (talk) 04:43, 22 May 2010 (UTC)

Yes, indeed so. Brews ohare (talk) 04:50, 22 May 2010 (UTC)

Look at any article that Brews has edited. That's how we works; it's why it's so stressful to try to collaborate with him on an article. Dicklyon (talk) 20:13, 22 May 2010 (UTC)

Dick: A count of my edits tells nothing about either (i) whether these edits overall were constructive and added to the article, or (ii) whether it was a nuisance for others that many edits were made. During this period of my activity the contributors were BobK31416 and myself, with occasional visits by Michael Hardy, David Tombe, and Carl (CBM). No-one complained about the number of my edits.

To judge their effectiveness, you might compare the article today with its condition when I began. Here are some of the notable differences:

1. Nine new figures drawn by myself.
2. Reorganization of topics such as Proof by Rearrangement where the text was uncoordinated with the figures and contained irrelevant technical gobbledygook like the Banach-Tarski paradox
3. Introduction of several new sections, such as General triangles using parallelograms, Inner product spaces, this last with assistance from Carl, and Lagrange's identity and Pythagoras' theorem in collaboration with David Tombe.
4. Rewrite and clarification of several sections such as Proof using differentials, which involved extensive collaboration with BobK, and Algebraic proof, Differential geometry.
5. Addition of 32 new citations.
6. Clean-up of many sections, such as History, Complex arithmetic, Euclid's proof.

These extensive changes naturally required many edits, but they were to good effect.

Your smear of my efforts is unwarranted and superficial. Most of your objections to my work stem from your own haste to jump to conclusions and your blithe indifference to all explanation. I would hope in future you might take the view that I am seriously trying to improve WP, and not aiming to annoy you. That change in your attitude would make it easier for us to collaborate. Brews ohare (talk) 23:38, 22 May 2010 (UTC)

I have never suggested that it's about me. And "blithe indifference to all explanation" is pot calling kettle black. My comments were about how you edit, not which edits were good and which bad; there are some of each in your expansion of the article from 40 KB to a bloated 70 KB. Dicklyon (talk) 06:48, 23 May 2010 (UTC)

Large numbers of edits, many of them unexplained, can be annoying in part because of a lack of transparency and peer-review. However, I find the article substantially improved over the 40 KB version of it from a month ago. If I understand rightly, then the objective is to bring the article up to FA status. This is a worthy goal, and should be the ultimate measuring rod with which we assess improvement. I think the next step will be to submit the article for peer review. While it would be helpful to get the opinion of a mathematician in this process, the input of a non-mathematician is probably equally desirable. Sławomir Biały (talk) 11:17, 23 May 2010 (UTC)

I agree. Given some things that happened on the talk page (not involving Brews), I took the complaint seriously and compared the current version with the version before Brews' first edit on this article. The article didn't have any really glaring gaps when he started, so obviously his additions weren't strictly necessary. But the current version is clearly better. Hans Adler 12:00, 23 May 2010 (UTC)

I didn't mean to imply that the article is not improved, as I have not studied the difference. But in respect to the question "Has it been so broken that it needs this flurry of activity to fix it?", the point is that the recent flurry of activity was not based on any property of the article, but on the editor. This needs to be understood, especially in light of the difficulty that it has caused as many editors tried to talk sense into Brews on one particular bad section. He was banned from editing any physics articles or talk pages for a year, based on similar behavior. It's sad to see that he has done nothing to reform the behaviors that wasted so much time of so many good editors. Dicklyon (talk) 16:38, 23 May 2010 (UTC)

This has come up again and again: I have pointed it out as have others, it came up in the arbitration case and in related actions. It don't know how many times it needs to be repeated before he takes notice and respects WPs norms of behaviour.--JohnBlackburne^words_deeds 21:18, 23 May 2010 (UTC)

John: [Finell's complaint was about making fine-scale adjustments on his Talk page, causing the nuisance of an automatic flag to appear that he had new messages on his talk page, when only minor rewording had happened. That behavior has been corrected, and has nothing to do with the edits here. Brews ohare (talk) 02:36, 24 May 2010 (UTC)

The multitude of edits makes me not want to investigate what changed. It requires deciding how far back to set the "Compare selected versions" feature on the history page. Admittedly not hard to do, but still an inconvenience. Pressing the 'diff' link on my watch page may only tell me what the same editor changed five minutes ago. If there are intermediate changes by other editors, it requires multiple uses of the Compare feature (spanning each cluster of edits) to determine who changed what.

Something to consider: So many edits within minutes of each other by one editor may give one or more of these first impressions:

• The edits may contain half-baked ideas because the editor apparently changed their mind (again) just minutes after their last edit of the same seciton, or
• A zealot found a new object to obsess on; interfering with the "mission" of someone who is fixated isn't likely to have a happy outcome.

Not saying that those would be accurate impressions in this case, but those are the first things that come to mind when I see one editor filling up an edit history.

Perhaps there is the concern that someone else may jump in with an idea first, so it is necessary to update the page as each new thought comes to mind? Is it so important to make the change before someone else can?

I suggest tweaking sections of the text in a sandbox on a user page and making less frequent changes to the article that is on the watch list of more than 200 other editors:

http://toolserver.org/~mzmcbride/watcher/?db=enwiki_p&titles=Pythagorean_theorem

-Ac44ck (talk) 04:04, 24 May 2010 (UTC)

The sandbox is a good suggestion that I should use too, but perhaps you should be more careful not to make personal attacks and not to speculate about what might be in another editor's mind. I think any suggestions from you on edits for the article that would improve it would be welcomed. --Bob K31416 (talk) 04:23, 24 May 2010 (UTC)

I'd say that the vast majority of edits made by any editor are small edits, like correcting a sentence, adding a source, rearranging a few sentences. That is what my edits are too. If I had twenty user names and did exactly what has been done here under twenty names, nobody would give a &%$*. That is particularly true if they had no idea who I was, and could put aside past differences Brews ohare (talk) 05:08, 24 May 2010 (UTC)

I had no idea who you are. There were no past differences. I just saw a vast number of edits in a short time. For me, it makes it not worth the effort to keep up with what is going on because it's probably going to change again in ten minutes. If twenty editors made the same number of edits in the same time frame, my first impression would be that someone had started an edit war. There have been more times times than I would like to admit where I changed my edits a number of times in a short period, but not day after day. In my case, I hadn't thought through I wanted to write before pressing the 'save' button. Perhaps that is not the case here. In my experience, tt is an anomalous editing practice to change an article so frequently. - Ac44ck (talk) 11:42, 24 May 2010 (UTC)

I am not sure what the problem is unless you own the article. Brews is obviously not going to vandalise, and any regressions he might inadvertently introduce are unlikely to be noticed by the article's main audience. While he is so active we also needn't worry much about vandalism being added unnoticed. Just do it like me: Ignore the individual edits that appear on your watchlist and make an occasional long-distance diff to see what has changed over the last week or so. Hans Adler 12:01, 24 May 2010 (UTC)

The problem is that when Brews adopts an article, he wants to own it, and pretty much doesn't allow others to stop him when what he's adding is not helpful, or is nonsensical. He pissed off enough editors this way that we was given a one-year ban or editing all physics topics, broadly construed, as well as any talk page discussions about physics topics. One doesn't bring down that kind of reaction from arbcom by being a sensible editor; maybe he is less bad now, but recent behavior here, on the whacky pentagon section, suggests that he has learned little, if anything, about how to collaborate with other editors. Dicklyon (talk) 02:26, 26 May 2010 (UTC)

Dick: Your interpretation of past history, and your dragging up nonexistent difficulties with this article on Pythagoras' theorem, reflects more upon yourself than upon me. Brews ohare (talk) 03:38, 26 May 2010 (UTC)

The "nonexistent difficulties" started when I commented on your taking the problem to golden ratio, in Talk:Golden ratio#New big mathy section. Since then, we've got about 100 KB of discussion about it here (see #Pentagon and #Deletion of side of pentagon above, among others), with you on one side, and a half dozen others on the other side. Business as usual. Dicklyon (talk) 04:15, 26 May 2010 (UTC)

No, that's not how it is, or was. Mountains from molehills. Brews ohare (talk) 05:13, 26 May 2010 (UTC)

Three weeks and 165 edits by Brews later, expanding the article from a bloated 69K to a really plump 85K, it now looks like this:

569 (351/218) Brews ohare  	2010-04-24 15:25  	2010-06-11 16:40
216 (138/78)  Michael Hardy 	2003-03-26 02:30 	2010-05-09 03:00
90 (87/3)     Bob K31416 	2010-05-01 13:07 	2010-05-26 12:22

I guess it was pretty bad before. Dicklyon (talk) 04:29, 15 June 2010 (UTC)

Edits are for improvement of the article, not implied criticism of what was there before. So I'd suggest the various changes indicate opportunities taken to make the article better. Some changes improved documentation (sources), some improved organization, some improved or added figures, some improved examples or their discussion. Brews ohare (talk) 19:45, 16 June 2010 (UTC)

Pythagoras Made Difficult

The suggested source for Michael Hardy's differential proof of Pythagoras' theorem ‘Pythagoras Made Difficult’ appears to be erroneous. The Table of contents for the suggested issue of the Mathematical Intelligencer has no such listing at any page number. Brews ohare (talk) 15:51, 15 June 2010 (UTC)

No, the thing being referenced does exist, and the reference is correct. That sort of publication often has little half-page things, and the official "table of contents" often lumps them in with the previous longer article. Actually, the previous article ends at the bottom of page 29. It's crucial to actually check references for this sort of reason. — Carl (CBM · talk) 16:13, 15 June 2010 (UTC)

Hi Carl: I did check this matter as far as possible: it appears nowhere in the search engine for the journal. As the Table of contents link shows, it appears nowhere in the table of contents. Google Scholar turns up no publication whatsoever. The article that is listed as ending on p. 31 is available only by purchase or request through a library, so it is very difficult to determine whether there is some kind of tag-on as you suggest.

Google search turns up WP, which refers to this source. A reference without volume or page is found in Proof #40. In sum, there appears to be no way to find this source other than WP. Perhaps you can illuminate just how to track this reference down aside from taking a flier and purchasing the document on Ramanujan ending on page 31? Brews ohare (talk) 16:44, 15 June 2010 (UTC)

The standard way to verify a source with certainty and no cost is to walk to the library, open the journal being referenced, and flip it to the desired page. When that's impractical, I think the best practice on WP is to ask for someone at the math project to verify it; many people will have free access to that journal at their institution. In this specific case, you could also ask User:Michael Hardy. Personally, I feel that "this citation is fictitious" is a very strong claim to make without looking at the journal itself.

But there's no harm done, and I am physically looking at the source as I write this. It's an illustrated half-page note about using differential calculus to prove the Pythagorean theorem. — Carl (CBM · talk) 17:01, 15 June 2010 (UTC)

Carl: By saying the citation was erroneous, I did not mean that it was ‘fictitious’ or "made up". I meant that it didn't appear in the journal's own search engine or table of contents, so perhaps the article was from another location. However, another point that could be made is that this article is hard to find, and the exact content may or may not be portrayed accurately in the summary at Proof #40, which is probably what people will find. That summary is wanting in several respects, leading to the suggestion in that source that “It must be taken with a grain of salt.” As I pointed out here, the summary of Hardy's proof leaves some gaps. On the other hand, Staring's proof seems to be air-tight. It would be nice to know if Hardy's is as well. Brews ohare (talk) 17:19, 15 June 2010 (UTC)

BTW, Carl, a "walk to the library" won't do it for me. Neither the U of A library nor the U of NV library carry this back issue from 1988, and it would be ordered from a remote site upon my request as a faculty member. I don't think that kind of "access" is at all useful to readers of WP, do you? Brews ohare (talk) 17:27, 15 June 2010 (UTC)

But this source is not hard to find: it's exactly where the reference says it is, it just isn't online. There's no reason that our sources have to be online. The U of NV library page says that v. 10 is available at "UNLV LASR, 1st Floor / QA1 .M427 v.10 1988". Even if it had to come from a remote site, that should only add a few days to the request. As for the proofs, they use exactly the same method. — Carl (CBM · talk) 17:51, 15 June 2010 (UTC)

Hi Carl: Well, to backtrack a bit, I thought the invisibility of this source to Google, Google scholar and to the Journal's own ToC and search engine, indicated an incorrect listing that should be fixed. Now I am reassured by yourself that it really exists, and it is to be found as indicated. Great. I suspect my doubts would be aroused in most WP readers who are inclined to wonder about accuracy issues. So I've added a little reassurance to the listing.

I'd also suggest to you that although Hardy's proof is based upon the same idea of differential changes in the sides, that is where the resemblance to Staring's proof stops. Maybe you'd comment on my attempt to counter Proof 40's objections to Hardy's proof. Unfortunately they were met with claims of OR. Brews ohare (talk) 18:20, 15 June 2010 (UTC)

Are you sure that it is not available, if you are faculty at a research institution, check with your library to see whether you have online access to it. You mentioned, "my request as a faculty member", so I gather you are faculty. If it isn't available at your library either online or in dead-tree version, check whether you can get it through inter-library loan. Librarians can be very helpful, ask them, they may have other suggestions. TomS TDotO (talk) 18:16, 15 June 2010 (UTC)

Of course I can obtain it by inter-library loan. And yes, it is not available on line even through the U. However, please read my response to Carl above. Brews ohare (talk) 18:20, 15 June 2010 (UTC)

I'd suggest that addition of this reference to a relatively “invisible” source (a source that is not identified on the Journal's own web pages, nor by searches using Google and Google Scholar, and available only to those willing to believe it exists because WP says so, and being so motivated are willing to search for it on faith through inter-library loan as existing as an “invisible” appendage to another unrelated article), such a reference serves a doubtful purpose here. Brews ohare (talk) 00:26, 17 June 2010 (UTC)

Pending changes

This article is one of a number (about 100) selected for the early stage of the trial of the Wikipedia:Pending Changes system on the English language Wikipedia. All the articles listed at Wikipedia:Pending changes/Queue are being considered for level 1 pending changes protection.

The following request appears on that page:

Comments on the suitability of theis page for "Penfding changes" would be appreciated.

Please update the Queue page as appropriate.

Note that I am not involved in this project any much more than any other editor, just posting these notes since it is quite a big change, potentially

Regards, Rich Farmbrough, 23:37, 16 June 2010 (UTC).

Note that, despite this message, the present article is not scheduled to be part of the trial at the moment, although it is on a list of articles that someone proposed. — Carl (CBM · talk) 00:31, 17 June 2010 (UTC)

Inner product spaces

I undid a change that added a "Euclidean" restriction; the point of the section is that the generalization works for arbitrary inner product spaces. "Euclidean inner product spaces", if read in the natural way, is a synonym for "Euclidean spaces", i.e. spaces of the form $\mathbb {R} ^{n}$ and their isomorphic copies.

Much of the current emphasis of the second paragraph of that section is misplaced. In an inner product space, the norm that matters is the one that comes from the inner product. It would be exceedingly rare to look at an inner product space qua inner product space, while simultaneously using a different norm to generate the topology. So the info about the parallelogram inequality and arbitrary norms is not particularly relevant. — Carl (CBM · talk) 18:30, 20 June 2010 (UTC)

Hi Carl: From what I can see, in the literature on functional analysis there is some ambiguity in the use of the term "Euclidean space". For example, Prugovec̆ki (§2.1) uses Euclidean space as synonymous with a vector space using an inner product. On that basis the designation "Euclidean" imposes no restriction upon the "general" inner product space: they are the same thing. Other authors appear to use the term Euclidean space only for an inner product space using the Euclidean ("natural") norm. You suggest that a more common usage is for $\mathbb {R} ^{n}$. In view of this dichotomy in the literature, how would you word the differences, and how would you explain how $\mathbb {R} ^{n}$ is different from a vector space with an inner product? Getting terminology straight is a great aid to the reader trying to understand various books and WP articles.

The distinction between an inner product space with some peculiar norm not the "natural" norm and the inner product space with the natural norm seemed necessary because the idea of norm is more general than the natural norm. Too much may have been said in the attempt to get the point across. The article on inner product spaces should be improved to discuss this issue and relieve the need for much discussion here. At the moment, I have added a sentence or two and some sources to inner product spaces as a beginning to widen its scope, but more should be done.

I am concerned that the article should not present Pythagoras' theorem as some kind of additional postulate for Euclidean spaces with the inner product norm, but clearly show it is a derived result from the postulates on inner products.

Perhaps you could propose a different organization of this section that would make the reader aware of oversimplifications without dwelling upon them? Brews ohare (talk) 00:20, 21 June 2010 (UTC)

We can just say "finite-dimensional Euclidean space" if you prefer. I don't think it's necessary to try to explain the difference here; that's out of scope for this article. Neither do we need to explain what a norm is, we can just link to that article. — Carl (CBM · talk) 00:24, 21 June 2010 (UTC)

Carl: A space of Legendre polynomials of degree up to n is Euclidean in some "normal usage" while one with infinite n is not? Can you elaborate upon how reference to "finite dimensional Euclidean space" will help matters?

Reference to normed space isn't useful with that article in the shape that it is in. Context has to be provided. Brews ohare (talk) 01:11, 21 June 2010 (UTC)

(1) It seems likely to me that virtually every author's "finite dimensional Euclidean space" will be isomorphic to $\mathbb {R} ^{n}$ for some n. But this article really is not the place to dwell on the definition of a Euclidean space. We can just say "spaces of the form $\mathbb {R} ^{n}$" if you really prefer that.

(2) I do not believe that the bad state of other articles is a reason to expand this article beyond its scope. Otherwise, for example, we would end up duplicating material everywhere when we link to a stub. We should make links as if the destination article is basically complete. However, normed vector space is not in extremely bad shape anyway. — Carl (CBM · talk) 01:18, 21 June 2010 (UTC)

Carl: Point (1): I don't understand the difficulty with the present wording. Can you please tell me in detail what you are looking for and what objections are to be met? I do not understand what the "spaces of the type $\mathbb {R} ^{n}$" is supposed to convey that is different from what is presently conveyed. Is the idea that dot products are to replace inner products, or what? That replacement would kill the whole point of the subsection.

Point (2): Yes, duplication could happen. Here are problems I see. The general article is written from a perspective remote from the application here. Hence, some context helping the reader to understand how the general article works in this context is needed. Another problem: The general article is formulated (in principle) to present the topic as a stand-alone, and as such must introduce matters in a general way appropriate to that broad topic, involving notions and development for this topic appropriate in the abstract. However, here we need only a very limited directed subset of ideas, organized for this application. Third problem: The general article is in fact not presented in an appropriate manner at the moment, for even its own objectives, and may stay that way for a decade unless somebody gets motivated to do something about it. There is no need to cast this shadow over this Pythagoras article when a paragraph here can fix matters. It really is a question of just how much space is used, and can it be afforded. Do you agree? Brews ohare (talk) 02:59, 21 June 2010 (UTC)

No, I don't agree. The article normed vector space is not in that bad of shape. It has a basic outline of the topic and a definition. It does not cast any sort of "shadow" over this article. This is not a textbook, it's a reference work, and we do not need to go into depth establishing context for inner product spaces, which are already a secondary point in this article. I think that even if we had space, it is off-topic here to go into depth about the parallelogram inequality.

As for the term "Euclidean space", the article here was using it in a way quite different than our article Euclidean space, which does include the restriction to finite-dimensional spaces, and in a way different than most mathematicians would expect. But the present revision of the article [2] is fine. — Carl (CBM · talk) 03:08, 21 June 2010 (UTC)

Separate problem

There is a subtle problem in this text:

Distance can be introduced in an inner product space, making it a normed inner product space by introducing the norm, denoted ||v|| for a vector v. A norm can be defined in a variety of ways, but a norm is expressible as an inner product if and only if the parallelogram equality holds:

$2\|\mathbf {u} \|^{2}+2\|\mathbf {v} \|^{2}=\|\mathbf {u} +\mathbf {v} \|^{2}+\|\mathbf {u} -\mathbf {v} \|^{2}\ ,$

for every pair u and v in the space. This requirement is satisfied only by the norm ||v||² = <v,v>, often called the Euclidean norm, and such a space often is called a Euclidean space.

The text goes from telling how to define a norm on an inner product space to telling how to test whether a norm came from an inner product. This leads to the error: there could be other norms on the space that also satisfy the parallelogram inequality, which arise from other inner products on the space. So the "only by the norm" claim is wrong. The other norms arising from other inner products would not be relevant to the inner product space at hand, though. — Carl (CBM · talk) 01:30, 21 June 2010 (UTC)

Carl: Nothing in what is said restricts $\langle u,v\rangle \,$ to a unique form for the norm: any definition of inner product $\langle u,v\rangle \,$ (and they may be legion) satisfying the axioms for inner product is OK. That agrees 100% with what you have said: “there could be other norms on the space that also satisfy the parallelogram inequality, which arise from other inner products on the space”. I think the text has been misread. The parallelogram inequality is mentioned only for the reasons you mention, as a test: given a norm, can it be associated with an inner product? The answer is: “not unless it satisfies the parallelogram equality”. Again, there is no indication that one and only one norm will pass the test; the claim is that only an inner product norm will pass the test, and that can be any inner product norm. Perhpas you could suggest some wording that would avid misreading what is meant? Brews ohare (talk) 02:41, 21 June 2010 (UTC)

But an inner product space comes with a particular inner product, and that is the one that the notation $\langle v,v\rangle$ refers to. So the text claims that every norm on the space that satisfies the parallelogram inequality comes from the initially-given inner product on the space. This isn't true. — Carl (CBM · talk) 02:51, 21 June 2010 (UTC)

I'll try some rewording to avoid that impression. Give me a little time to do that. Brews ohare (talk) 03:02, 21 June 2010 (UTC)

I don't think we particularly need to mention the parallelogram inequality, or arbitrary norms, in the first place. — Carl (CBM · talk) 03:09, 21 June 2010 (UTC)

I see that you have revised this section already. I don't see that your revision addresses the issue you raise of multiple possible choices for the inner product used for the norm, of which the inner product defining the space is only one of legion. Brews ohare (talk) 03:12, 21 June 2010 (UTC)

The space comes with an inner product, denoted $\langle v,w\rangle$. The norm on the space is defined as $||v||={\sqrt {\langle v,v\rangle }}$. Here the inner product is obviously the same as the one used for the space, because the same notation is used for it. I'm not really sure what you're asking. — Carl (CBM · talk) 03:16, 21 June 2010 (UTC)

Carl: I may be off the beam here. What I'm thinking is that hypothetically I could define an inner product in several ways. I don't know if this would work, but suppose I introduce into the vector space several inner products:

$\langle u,v\rangle _{j}=\int m_{j}(x)u(x)v(x)\ dx$

where m_j is some weighting function. Assuming the required axioms are satisfied, I use j=1 for the space inner product defining the space, and j=2 for the norm of this space based upon a second inner product. The norm satisfies the parallelogram equality. There may be some difficulties with orthogonality, I don't know. Maybe a different approach is needed to achieve the goal. But you get the point. Brews ohare (talk) 03:29, 21 June 2010 (UTC)

An inner product space comes with an inner product. If you change the inner product you have a different inner product space. The norm we care here about is the one that comes from the inner product associated with the space, because the Pythagorean theorem is about the norm of the sum of orthogonal vectors, and the "norm" there has to match the "orthogonal" condition. But we do not need to go into depth about this in the article. We can just start with the inner product, define the norm from it, and then state the Pythagorean theorem. — Carl (CBM · talk) 03:31, 21 June 2010 (UTC)

I think you are asserting that there is a theorem somewhere that two inner product spaces based upon the same vector space cannot be found such that the inner product of inner product space 1 can be used as the norm of space 2? Brews ohare (talk) 03:38, 21 June 2010 (UTC)

I don't understand what you're asking. The point is that an inner product space starts with a fixed inner product, and then you define the norm from that. Indeed, many people consider an inner product space to be a special case of a normed vector space (just as Hilbert spaces are special cases of Banach spaces). — Carl (CBM · talk) 03:41, 21 June 2010 (UTC)

If you are asking whether two different inner products can give rise to the same norm, the answer is no; see polarization identity. Again, this is beyond the scope of the present article, for which inner product spaces are already a secondary point. — Carl (CBM · talk) 03:47, 21 June 2010 (UTC)

Carl: I am asking whether two different inner products can give rise to two different norms, but both norms are viable in conjunction with one of the two inner products. That is, there are two or more norms possible, and the norm is not restricted to only the norm defined by the inner product defining the space. Brews ohare (talk) 04:15, 21 June 2010 (UTC)

I still don't understand what you're asking. What do you mean by "viable"? Each norm can be compatible with at most one inner product, because the polarization identity lets you recover the inner product from the norm as long as the norm actually does come from an inner product. — Carl (CBM · talk) 04:19, 21 June 2010 (UTC)

Thanks, Carl. That is the answer I was looking for. I wasn't aware that the inner product could be recovered like that. You'll notice that Inner product space doesn't mention the polarization identity. So the bottom line here is that there are many norms available other than the inner product norm, but they don't satisfy the parallelogram identity. I guess one could ask what is so special that we want to focus only on the inner product norm, and forget about the multitude of other norms that presumably were introduced for a purpose? My guess is that they don't satisfy Pythagoras' theorem, and so are non-Euclidean spaces? Brews ohare (talk) 04:38, 21 June 2010 (UTC)

No, any norm that arises from an inner product satisfies the parallelogram identity. The point is that the choice has already been made when you are presented with an inner product space. If choose a different inner product, you get a different inner product space. — Carl (CBM · talk) 11:00, 21 June 2010 (UTC)

Carl: We're a bit on different wavelengths here. I do understand what you have said above. My curiosity is about normed spaces that are not inner product spaces. One can still introduce a distance between u and v as d =||u−v||. Does Pythagoras' play a role in such spaces? Brews ohare (talk) 15:03, 21 June 2010 (UTC)

OK, I see. The statement of the Pythagorean theorem requires both distance (norm) and angle (orthogonality). In an normed space that is not an inner product space, there is a concept of distance but no concept of angle. So it doesn't make much sense to talk about the Pythagorean theorem in that context. — Carl (CBM · talk) 15:05, 21 June 2010 (UTC)

Thanks Carl. A further question: Is there any analog to (say) hyperbolic or spherical geometry in function spaces? In other words, can we introduce replacements for Pythagoras in a vector space like :$\cosh \|c\|=\cosh \|a\|\ \cosh \|b\|-\sinh \|a\|\ \sinh \|b\|\ \cos \gamma \,$? Thanks for your help. Brews ohare (talk) 15:21, 21 June 2010 (UTC)

That still refers to an angle. There simply is no concept of angle in a general normed vector space. — Carl (CBM · talk) 15:54, 21 June 2010 (UTC)

Sorry, I wasn't clear. The question is: how to introduce angle into an n-dimensional vector space without implying Pythagoras? Apparently that requires a norm that is not an inner product norm, as the inner product norm apparently requires Pythagoras. It can be done in spherical and hyperbolic geometry - is there an analog? Thank you again. Brews ohare (talk) 18:42, 23 June 2010 (UTC)

The way you introduce angles is by giving the space an inner product. In a vector space all the vectors originate at the origin. So the concept of angle on the surface of a sphere does not seem to be related to the concept of angle in a vector space. — Carl (CBM · talk) 19:23, 23 June 2010 (UTC)

I see no reason to ask "how to introduce angle into an n-dimensional vector space without implying Pythagoras" on the talk page for Pythagoras' theorem. You might consider taking this to the Wikipedia:Reference_desk/Mathematics. DVdm (talk) 19:27, 23 June 2010 (UTC)

DVdm: The point of asking is to determine whether there is a body of knowledge as yet unnoticed in Pythagorean theorem that should get a mention. Possibly Wikipedia:Reference_desk/Mathematics could answer the question, but I thought Carl was likely to know. Brews ohare (talk) 20:40, 23 June 2010 (UTC)