Definition talk:Euclidean Space

It occurred to me that this page already exists, as a Proof (as opposed to a Definition).

If this page is just redirected to the Proof page, would that make the definition count wrong? Does it matter? --Matt Westwood 11:20, 11 January 2009 (UTC)
 * I'm going to reply on your talk page, since this article is already redirected. Zelmerszoetrop 13:06, 11 January 2009 (UTC)

I was hasty - I've un-redirected the original definition page. I'll leave this be for the moment and get on with all that tedious analysis stuff ... --Matt Westwood 14:25, 11 January 2009 (UTC)

Definition properties
At base, the definition that I have is that a "Euclidean space" is the metric space formed by the vector space $\R^n$ with the Euclidean metric on it. The other properties on top (addition, scalar multiplication, all that stuff) follow as properties of that structure. Therefore there is no need to state them as necessary properties of this object. If it is considered necessary to alert the user to all these properties immediately on hitting this page, then IMO they should be indicated in the "Also see" section.

Or am I wrong, and that it is necessary to define all these properties up front (i.e. they do not naturally follow from the properties of the vector space $\R^n$ and need to be stated separately) then I think I need to go back to school. --prime mover 02:15, 18 March 2012 (EDT)


 * Well, there is some kind of issue here. Do we start from the dot product and define the Euclidean norm in terms of the dot product, or do we start from the Euclidean norm and define the dot product of two vectors in terms of the Euclidean norm and the angle between the two vectors? Abcxyz 10:22, 18 March 2012 (EDT)


 * I don't see the problem. We define a vector space. Then we define the Euclidean space as a (particular) vector space with a (particular) metric imposed. Then we define a dot product from the first definition. We sideline the "alternative definition" to a proof page somewhere. We define the Euclidean norm as it is defined.
 * The only problem is the definition of the dot product which needs to be proved as a result rather than cited as a definition. The fact that in applied maths the nature of Euclidean space is taken for granted means that it is in those cases feasible to define it in terms of angles.
 * That's the way I see it. Is that feasible or are there still circularities? --prime mover 10:34, 18 March 2012 (EDT)


 * Exactly what do you mean by the "first definition" of the dot product? Abcxyz 10:36, 18 March 2012 (EDT)


 * He means $\sum_{i=1}^{n} a_i b_i$ --GFauxPas 10:44, 18 March 2012 (EDT)
 * I think it comes down to how you define "the angle between two vectors", which both Fraleigh and Larson define using $\arccos \dfrac {\mathbf {v \cdot w}}{\left\Vert{\mathbf v}\right\Vert \left\Vert{\mathbf w}\right\Vert}$. I'll poke around and see if I can find another def'n of the angle between two vectors. --GFauxPas 11:00, 18 March 2012 (EDT)
 * Khanhas a different definition. He defines, if you don't want to watch the video:


 * Let $\mathbf a$ and $\mathbf b$ be non zero.


 * Draw a 2D triangle with sides of length $\left\Vert{\mathbf a}\right\Vert, \left\Vert{\mathbf b}\right\Vert, \left\Vert{\mathbf {a - b} }\right\Vert$. (triangle inequality guarantees that such a triangle is defined).


 * The angle between $\mathbf a$ and $\mathbf b$ := the angle between sides $\left\Vert{\mathbf a}\right\Vert$ and $\left\Vert{\mathbf b}\right\Vert$ in the corresponding triangle.


 * Interestingly enough, according to Khan's approach, the second definition of dot product is a theorem anyway, and the proof comes from Law of Cosines!. So it either solves our problems or makes them worse, depending on how you look at it. --GFauxPas 11:36, 18 March 2012 (EDT)


 * The whole point is: I don't think we can define the dot product by the cosine rule as we have this currently set up - best we can do is, haveing established facts about the Euclidean space, is prove dot product has this property. --prime mover 11:39, 18 March 2012 (EDT)

The impression I'm getting is that prime mover is defining the structures of Euclidean space (that is, dot product, Euclidean norm, Euclidean metric, etc.) pretty much independently. Is that right, or am I mistaken somewhere? Abcxyz 11:45, 18 March 2012 (EDT)


 * Not sure what you mean by "independently". We just don't a definition to be circular. --GFauxPas 12:39, 18 March 2012 (EDT)


 * I mean, for example, that (as prime mover does it) the definition of Euclidean norm does not invoke the definition of the dot product. Abcxyz 12:45, 18 March 2012 (EDT)


 * Note that the second definition of dot product is defined using the Euclidean norm, though. --GFauxPas 12:55, 18 March 2012 (EDT)


 * Yes, I know. But prime mover doesn't define the dot product using the Euclidean norm. Abcxyz 12:58, 18 March 2012 (EDT)