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Often it is said that Euclidean $n$-dimensional space over $\mathbb{R}$ is different from $\mathbb{R}^n$, because in Euclidean space, all points are equivalent, in $\mathbb{R}^n$, there is origin.

When I just heard this first time, I thought that the point $0$ is different from others in $\mathbb{R}^n$ when one concerns algebra: it is identity element (of a group/vector space/...). Otherwise,...possibly no problem. Why it is not correct to say $\mathbb{R}^n$ as Euclidean space?

Some books on Geometry also have some discussion or remarks on such topic (see this).

Even in some of my lectures, when I wrote

Let $\mathbb{R}^n$ denotes Euclidean space...,

some experts (from Geometry) said ''don't say $\mathbb{R}^n$ (or $\mathbb{R}^n, \langle\,\, ,\,\,\rangle$)as Euclidean space''.

Is it not correct to teach Euclidean geometry by starting like this-

Consider vector space (set) $\mathbb{R}^n$ with dot product.

We say it is Euclidean space.

Points in it are $n$-tuples.

Lines are cosets of $1$-dimensional subspaces $x+W$.

Two lines $x+W$ and $y+W'$ are orthogonal if $W,W'$ are orthogonal.

(and so on, ....). Whats serious problems come if we teach geometry like this?

Groups
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  • If the last section is not correct, can we start as Without loss of generality, let $\mathbb{R}^n$ with dot product denote Euclidean $n$-space. Points in it are ..... – Groups Nov 19 '15 at 10:16

1 Answers1

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You can see it in this way: it's a matter of choosing coordinate system. Formally, an Euclidean space $\bf{E}$ is a metric vector space; when we want to make computations over $\bf{E}$, we may like to choose a basis for the underlying vector space. Or, in other words, we are using the (non-canonical) isomorphism $\mathbf{E}\simeq\mathbb{R}^n$.

Saying that $\mathbb{R}^n$ is an Euclidean space is not completely wrong, it only assumes that you have chosen a basis for the underlying vector space.

Caligula
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  • So what would be appropriate word for $\mathbb{R}^n$ which includes Euclidean? Possibly can we say - Let $\mathbb{R}^n$ denote n-dim. affine Euclidean space or let $\mathbb{R}^n$ denote (n-dim. affine Euclidean space)+(coordinates)? – Groups Nov 19 '15 at 10:49
  • Usually I think of $\mathbb{R}^n$ as a vector space "with euclidean [or also metric] structure" when I want to emphasize the fact that I'm using it as representation of euclidean space. I think, perhaps, that it's a bit sophistic to note such details, unless you're working on something where the difference is crucial. – Caligula Nov 19 '15 at 10:53