Euclidean spaces

Ingredients

Concepts

Constructions

Examples

Theorems

# Euclidean spaces

## Idea

The concept of Euclidean space in analysis, topology, differential geometry and specifically Euclidean geometry, and physics is a fomalization in modern terms of the spaces studied in Euclid 300BC, equipped with the structures that Euclid recognised his spaces as having.

In the strict sense of the word, Euclidean space $E^n$ of dimension $n$ is, up to isometry, the metric space whose underlying set is the Cartesian space $\mathbb{R}^n$ and whose distance function $d$ is given by the Euclidean norm:

$d_{Eucl}(x,y) \coloneqq {\Vert x-y\Vert} = \sqrt{ \sum_{i = 1}^n (y_i - x_i)^2 } \,.$

In Euclid 300BC this is considered for $n = 3$; and it is considered not in terms of coordinate functions as above, but via axioms of synthetic geometry.

This means that in a Euclidean space one may construct for instance the unit sphere around any point, or the shortest curve connecting any two points. These are the operations studied in (Euclid 300BC), see at Euclidean geometry.

Of course these operations may be considered in every (other) metric space, too, see at non-Euclidean geometry. Euclidean geometry is distinguished notably from elliptic geometry or hyperbolic geometry by the fact that it satisfies the parallel postulate.

In regarding $E^n = (\mathbb{R}^n, d_{Eucl})$ (only) as a metric space, some extra structure still carried by $\mathbb{R}^n$ is disregarded, such as its vector space structure, hence its affine space structure and its canonical inner product space structure. Sometimes “Euclidean space” is used to refer to $E^n$ with that further extra structure remembered, which might then be called Cartesian space.

Retaining the inner product on top of the metric space structure means that on top of distances one may also speak of angles in a Euclidean space.

Then of course $\mathbb{R}^n$ carries also non-canonical inner product space structures, not corresponding to the Euclidean norm. Regarding $E^n$ as equipped with these one says that it is a pseudo-Euclidean space. These are now, again in the sense of Cartan geometry, the local model spaces for pseudo-Riemannian geometry.

Finally one could generalize and allow the dimension to be countably infinite, and regard separable Hilbert spaces as generalized Euclidean spaces.

## 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 $\mathbb{R}^n$ for $n$ the dimension. But Euclid's spaces had no coordinates; and in any case, what we do with them is still coordinate-independent.

## Euclidean spaces with infinitesimals

Instead of working in the real numbers $\mathbb{R}$ and $n$-dimensional real vector spaces $V$, one could instead work in a Archimedean ordered Artinian local $\mathbb{R}$-algebra $A$ and rank $n$ $A$-modules $V$. $A$ has infinitesimals, and so the $A$-modules $V$ have infinitesimals as well. Nevertheless, it is still possible to define the Euclidean distance function on $V$; the only difference is that the distance function is a pseudometric rather than a metric here.

Since $A$ is an local ring, the quotient of $A$ by its ideal of non-invertible elements $I$ is $\mathbb{R}$ itself, and the canonical function used in defining the quotient ring is the function $\Re:A \to \mathbb{R}$ which takes a number $a \in A$ to its purely real component $\Re(a) \in \mathbb{R}$. Since $A$ is an ordered $\mathbb{R}$-algebra, there is a strictly monotone ring homomorphism $h:\mathbb{R} \to A$.

The real numbers have lattice structure $\min:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and $\max:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$. This means that $A$ has a distance function given by the function $\rho:A \times A \to \mathbb{R}$, defined as

$\rho(a, b) \coloneqq \max(\Re(a), \Re(b)) - \min(\Re(a), \Re(b))$

as well as an absolute value given by the function $\vert-\vert:A \to \mathbb{R}$, defined as

$\vert a \vert \coloneqq \rho(a, 0)$

Since $\min(a, b) \leq \max(a, b)$, the pseudometric and multiplicative seminorm are always non-negative. In addition, by definition, the pseudometric takes any two elements $a \in A$ and $b \in A$ whose difference $a - b \in I$ is an infinitesimal to zero $\rho(a, b) = 0$.

Since $\mathbb{R}$ is an Euclidean field, it has a metric square root function $\sqrt{-}:[0, \infty) \to [0, \infty)$. Every rank $n$ $A$-module $V$ with basis $v:\mathrm{Fin}(n) \to V$ thus has a Euclidean pseudometric $\rho_V:V \times V \to K$ defined by

$\rho_V(a, b) \coloneqq \sqrt{\sum_{i \in \mathrm{Fin}(n)} \rho(a_i, b_i)^2}$

for module elements $a \in V$ and $b \in V$ and scalars $a_i \in A$ and $b_i \in A$ for index $i \in \mathrm{Fin}(n)$, where

$a = \sum_{i \in \mathrm{Fin}(n)} a_i v_i \quad b = \sum_{i \in \mathrm{Fin}(n)} b_i v_i$

If $A$ is an ordered field, then this reduces down to the Euclidean metric defined above.

## In constructive mathematics

In constructive mathematics, the real numbers used to define Euclidean spaces are the Dedekind real numbers $\mathbb{R}_{D}$, as those are the only ones that are Dedekind complete, in the sense of not having any gaps in the dense linear order. The Dedekind real numbers are also the real numbers that are geometrically contractible: whose shape is homotopically contractible $\esh(\mathbb{R}_D) \cong \mathbb{1}$.

### In predicative constructive mathematics

In predicative constructive mathematics, the Dedekind real numbers are defined relative to a universe $\mathcal{U}$, and thus there are many different such Dedekind real numbers that could be used to define Euclidean spaces, one $\mathbb{R}_\mathcal{U}$ for each $\mathcal{U}$. However, each set of Dedekind real numbers $\mathbb{R}_\mathcal{U}$ would be large relative to the sets in the universe $\mathcal{U}$.

If the predicative constructive foundations does not have universes, then there doesn’t exist any dense linear order that is actually Dedekind complete in the usual sense, and so the usual definition of Euclidean space does not work. Some mathematicians have proposed to use Sierpinski space $\Sigma$, the initial $\sigma$-frame, for defining the real numbers, in place of the large set of all propositions in a universe $\mathrm{Prop}_\mathcal{U}$, but the real numbers in that case are only $\Sigma$-Dedekind complete, which is a weaker condition than being Dedekind complete. Furthermore, Lešnik showed that for any two $\sigma$-frames $\Sigma$ and $\Sigma^{'}$ that embed into $\mathrm{Prop}_\mathcal{U}$ such that $\Sigma \subseteq \Sigma^{'}$, if $A$ is the $\Sigma$-Dedekind completion of $\mathbb{Q}$ and $B$ is the $\Sigma^{'}$-Dedekind completion of $\mathbb{Q}$, then $A \subseteq B$, so the $\Sigma$-Dedekind real numbers are not complete.

## 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

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

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

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

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

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 $\angle x z y$. In a $2$-dimensional Euclidean space, we can interpret $\angle x z y$ 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 $V$;

$\langle{x - z, y - z}\rangle = {\|\overline{x z}\|} {\|\overline{y z}\|} \cos \angle x z y ,$

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.

On the use of the Dedekind real numbers in constructive and predicative constructive mathematics, such as for Euclidean spaces: