is line an Euclidean space?

Question

On one hand, in order to be a Euclidean a space should be equipped with a concept of "an angle". And the angle on a line has only two values (0 and 180 degrees), while in higher dimensional spaces it can have a continuum of values. So, is it Euclidean? A simple test would be: are there theorems proven for a Euclidean space of arbitrary many dimensions, that fail in case of a 1D line?

Answer 1

$\mathbb{R}$ is indeed a Euclidean space.

Angles between nonzero vectors $x$ and $y$ in Euclidean space are defined by $$\theta = \arccos\dfrac{x\cdot y}{\| x\|\|y\|}\in [0,\pi].$$ In $\mathbb{R}$, $x\cdot y = xy$ and $\|x\|=|x|$, so necessarily $\dfrac{x\cdot y}{\| x\|\|y\|} = \dfrac{xy}{|xy|}=\pm 1$ and therefore these angles are necessarily either $\theta=0$ or $\theta=\pi$ as you mention. Nothing wrong with that.

Answer 2

A Euclidean space is a vector space $\mathbb R^n$ equipped with a scalar product that is positive definite. By putting $\langle x,y\rangle = xy$ in $\mathbb R$, you get your scalar product, so $(\mathbb R,\langle{}\cdot{},{}\cdot{}\rangle)$ is definitely Euclidean.