nLab
Euclidean space

A Euclidean space is an affine inner product space. That is, it is an affine space $E$ modelled on a vector space $V$ which has an inner product.

One generally takes the inner product to be positive-definite; otherwise, we say that $E$ is only a pseudo-Euclidean space. Also, one generally takes the dimension to be finite; Euclid? himself only considered dimensions up to $3$. For an infinite-dimensional Euclidean space, you would probably want $V$ to be a Hilbert space.

Arguably, the spaces studied by Euclid were not really modelled on inner product spaces, as the distances were lengths, not real numbers (which, if non-negative, are ratios of lengths). So we should say that $V$ has an inner product valued in some oriented line? $L$ (or rather, in $L^2$). Of course, Euclid did not use the inner product (which takes negative values) directly, but today we can recover it from what Euclid did discuss: lengths (valued in $L$) and angles (dimensionless).

Since the days of Rene Descartes?, it is common to identify a Euclidean space with a Cartesian space, that is ${ℝ}^n$ for $n$ the dimension. But Euclid's spaces had no coordinates; and in any case, what we do with them today is still coordinate-independent.

Lengths and angles

Given two points $x$ and $y$ of a Euclidean space $E$, their difference $x-y$ belongs to the vector space $V$, where it has a norm

$$\parallel x-y\parallel =\sqrt{⟨x-y,x-y⟩}.$$\|x - y\| = \sqrt{\langle{x - y, x - y}\rangle} .

This real number (or properly, element of the line $L$) is the distance between $x$ and $y$, or the length of the line segment $\overline{xy}$. This distance function makes $E$ into an ($L$-valued) metric space.

Given three points $x,y,z$, with $x,y\ne z$ (so that $\parallel x-z\parallel ,\parallel y-z\parallel \ne 0$), we can form the ratio

$$\frac{⟨x-z,y-z⟩}{\parallel x-z\parallel \parallel y-z\parallel },$$\frac{\langle{x - z, y - z}\rangle}{\|x - z\|\,\|y - z\|} ,

which is a (dimensionless) real number. By the Cauchy–Schwartz inequality, this number lies between $-1$ and $1$, so it's the cosine of a unique angle measure between $0$ and $\pi$ radians. This is the measure of the angle $\angle xzy$. In a $2$-dimensional Euclidean space, we can interpret $\angle xzy$ as a signed angle if we fix an orientation of $E$.