nLab
Euclidean space

Idea

A Euclidean space is a formulation in modern terms of the spaces that Euclid? studied, equipped with the structures that Euclid recognised his spaces as having.

Definition

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.

Remarks on terminology

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 René 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 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 (so taking values anywhere on the unit circle?) if we fix an orientation of $E$.

Conversely, knowing angles and lengths, we may recover the inner product on $V$;

$$⟨x-z,y-z⟩=\parallel \overline{xz}\parallel \parallel \overline{yz}\parallel \cos \angle xzy,$$\langle{x - z, y - z}\rangle = {\|\overline{x z}\|} {\|\overline{y z}\|} \cos \angle x z y ,

and other inner products are recovered by linearity. (We must then use the axioms of Euclidean geometry to prove that this is well defined and actually an inner product.) It’s actually possible to recover the inner product and angles from lengths alone; this is discussed at Hilbert space.