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I have some doubts about the precise meaning of Euclidean space. I understand Euclidean space as the

**metric space**[tex](\mathbb{R}^n,d)[/tex] where [tex]d[/tex] is the usual distance [tex]d(x,y)=\sqrt{\sum_i(x_i-y_i)^2}[/tex].

Now let's supose that we have our euclidean space (in 3D for simplicity) [tex](\mathbb{R}^3,d)[/tex] and then we consider the transformation to spherical coordinates. The "spherical coordinate space" [tex]\mathbb{R}^3[/tex] of 3-tuples [tex](r, \theta, \phi)[/tex], is itself an euclidean space?

As I see we have the choice to use also the distance [tex]d[/tex] on this latter space even if it has no physical meaning, but on the other hand it seems more useful to use the "phyisical" distance such that the distance between two points in the "spherical coordinate space" correspond to the physical distance between the points they represent. So, what distance to use? for each choice, is the space [tex]\mathbb{R}^3[/tex] of 3-tuples [tex](r, \theta, \phi)[/tex] also the euclidean space?