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?