Definition talk:Euclidean Space

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)

I hope I took prime mover's comments correctly. If not, please let me know. In the definition, when I wrote things such as "the Euclidean norm induced by the dot product", I only meant that the Euclidean norm is the norm induced by the inner product given by the dot product. I didn't mean that the Euclidean norm must be defined in terms of the dot product. There is certainly the perfectly valid approach of defining everything independently (that's what I think prime mover is doing, but I, of course, could be wrong). Would it be preferable to simply list the structures normally associated with Euclidean space (such as dot product, Euclidean norm, etc.) without all of the statements such as "the Euclidean metric is the metric induced by the Euclidean norm"? Abcxyz 00:20, 19 March 2012 (EDT)

For the record I have not contributed materially to either dot product or Euclidean norm. My only contributions in these areas have been cosmetic. As such, I have not been "defining these structures independently" as a deliberate act of malice. My particular concern is to ensure that all definitions are stated with as few premises as possible, without invoking circularity.

Specifically, and I'll try and say what I said above in different words (I fear I haven't made myself understood), I perceive a "Euclidean space" as being a real vector space with the Definition:Euclidean Metric on Real Vector Space superimposed on it. And as such that is all that is needed to define the Euclidean space. The facts about vector addition and scalar multiplication are consequences of the fact that we start with a vector space, so those do not need to be stated in the initial exposition.

Now, for the dot product stuff. The dot product as such is a function which takes as its arguments two vectors and outputs a scalar. And as such its definition is independent of the space upon which it is defined.

Now it just so happens that the Euclidean space, with the dot product imposed, is then an inner product space. This is a consequence of the construction and can be proved. It does not need to be invoked as part of the necessary definition for what a Euclidean space actually is.

(Oh and by the way can you think of an even more woolly and imprecise word than "considered"? I can't.)

Similarly it follows that the other properties are deduced from the already-established properties.

If you are seriously emotionally wedded to the idea of defining a Euclidean space in terms of these five properties, then there is a case for adding them as a separate section of this page, in a similar way to how the Axiom:Group Axioms are stated as the defining properties of a group, while a Definition:Group can also be described by building it up from simpler structures (magma, semigroup, monoid) and progressively adding further properties which restrict its nature more each time.

Except this case is not completely analogous, because in the case of the group axioms, the axioms do state the precise definition of a group, while in this case, the properties mentioned can be removed from the definition because they are, as I say, derived from the originally-stated properties from where this page started.

Unless of course I'm wrong, as is an overwhelmingly odds-on bet, in which case get one of the admins to block me from any further contributions to this site.

Does anyone else want to come in and help settle this one way or another? I fear it's taking up more of my time than I really have to spare. --prime mover 03:47, 19 March 2012 (EDT)

My elementary understanding of these kinds of things is that (generally) adding derivable things as definitions is useful only if (not "iff", "only if") they make concepts easier to understand. Here they don't. --GFauxPas 05:58, 19 March 2012 (EDT)

Please don't get me wrong. I sincerely apologize for having caused any confusion. I never thought prime mover was doing any "deliberate acts of malice". I never disagreed with prime mover. (By the way, when I said "defined", I only meant what prime mover posted on this talk page.) It was just that I did not yet understand what prime mover was getting at, and I asked questions to try to understand what prime mover was getting at. (I apologize for not getting the message before prime mover's last post.) Now that I think I understand much better, I still have the same question: Would it be preferable to just list the structures normally associated with Euclidean space, without all of the other stuff (such as stating that the Euclidean norm is the norm induced...)? Abcxyz 09:49, 19 March 2012 (EDT)

In my opinion, those have perfect right to be mentioned in the 'Also see' section. I'm not sure if I think they should be transcluded; the page would get a bit too big I fear. Also, please note that there is no such thing as a 'norm induced by a metric'; only the other way around. This is a quite intricate case, but I think it's best not to tire newcomers to the subject with the definition of a general inner product on this page, other than linked to separately via the definition of Dot product, etc.. --Lord_Farin 11:13, 19 March 2012 (EDT)

I also think the page would get too big if transclusions are included. (Anyway, I meant "norm induced by the dot product".) So we'll just revert the page to the previous version by prime mover and setup the "Also see" links? Abcxyz 11:38, 19 March 2012 (EDT)

That seems to be a sound course of action. --Lord_Farin 11:48, 19 March 2012 (EDT)

It's been done. By the way, regarding the use of the word "considered", I got the idea from other ProofWiki articles such as Definition:Topological Group. I didn't come up with that usage myself. Abcxyz 02:10, 20 March 2012 (EDT)