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It is well known that the Euclidean metric can be generalized to $\Bbb R^n$ by $\sqrt{(x_1-x'_1)^2+\cdots + (x_n-x'_n)^2}$, and that under this generalization it is still a metric and satisfies various other properties of the two and three dimensional cases.

But other than as a mathematical curiosity, why do this? Does this metric appear naturally in any mathematical problems or fields of study? Or is the only reason to generalize because we can?

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How do you really know that we do not live in 5-6 or n-dimensional space? So why stuck in 1-2-3-4 dimensional spaces and not generalize in the most natural way? – Metin Y. Apr 23 '13 at 21:39
The thing you have defined is a norm and not a metric. – Fly by Night Apr 23 '13 at 21:40
@FlybyNight Woops, you're right. – Jack M Apr 23 '13 at 21:42
Where to start...? Why work with the Euclidean norm? Because it comes from an inner product, which on top of that allows to identify $\mathbb{R}^n$ with its dual isometrically. Because it makes sense of the intuitive notion of "orthogonality". Because it is smooth away from $0$. Because it has nice and tractable isometries. Because... – 1015 Apr 23 '13 at 21:43
This norm also works in one dimensional case. – Integral Apr 23 '13 at 21:43

There are probably deeper answers to this, but one that comes to mind is that just because our world (high level physics considerations aside) is three dimensional, that doesn't mean that the only things that we want to measure in it are three dimensional. As an example let's assume you have a data set, say a point for $N$ different people and associated to each person you have a number $R$ of numerically-valued characteristics, e.g. height in centimeters, weight in kilograms etc. Then to each person we can associate a point in $R$ dimensional space, whose coordinates are the values of the characteristics.

Given this set of data we'd like to be able to tell whether two people are 'close' in some sense, so we need a metric, a notion of distance between the points. We already have the Euclidean metric in 3 dimensions, and we can generalise it to $R$ dimensions, so that's one notion of distance that we can choose. There are other metrics of course, many better suited, but this is an example of a situation where we need a metric on a space with more than 3 dimensions to tell us something about the real world.

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That would be an excellent example of what I'm talking about, but as you said, there's no reason why the Euclidean metric would be the best suited, there. The best metric would be determined by the specific problem - I personally have never found "it just sort of seems natural in a vague undefined way" to be a satisfactory motivation in mathematics. – Jack M Apr 24 '13 at 7:32

One important reason is just because we can. More than that, spaces of $n$ dimensions appear constantly in other fields of study like physics (even when they do not represent the physical space we live in, which seems to be 3 dimensional), and having that metric is usefull. For example phase spaces of a mechanical system have $2n$ dimensions, being $n$ the degrees of freedom. In quantum mechanics arbritarily high dimensions appear as Hilbert spaces are used, even infinite ones. Having a normed space is usefull when leading with these things.

More than that, we do it because we can, and I think that's the more important reason if we're talking about math. Also in math it leads to deeper results in other areas. There are more strange metric spaces with more strange distance functions that would ask about. But most of them are applied to something.

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