# On affine spaces, distances, angles, and coordinates

Let's define an affine space as a pair $(A, V)$, where $A$ is a set and $V$ is a vector space, together with a map $V\times A \rightarrow A, \;\; (v, a) \mapsto v + a,$ such that

1. $\forall \, a \in A: 0 + a = a\,$ ;
2. $\forall \, v, w \in V, \forall \, a \in A: v + (w + a) = (v + w) + a\,$ ;
3. $\forall \, a \in A$, the map $(v, a) \mapsto v + a\;$ is a bijection.

It is pretty clear that in an affine space defined in this way the concepts of distances and angles are not meaningful.

But now suppose that coordinates are introduced. Specifically, suppose that a an element $o$ of $A$ is chosen as the "origin", and some basis is chosen for $V$. For the time being, let's say that this basis is finite. Once an origin $o$ is chosen, the distinguished bijection $(v, o) \mapsto v + o$ can be used to assign to each $a \in A$, a unique $v \in V$ such that $a = v + o$, and once we specify a basis for $V$, this means that we can assign unique coordinates to each $a \in A$ (namely the coordinates of the unique vector $v$ such that $a = v + o$).

So now we have a (finite-dimensional) affine space with coordinates. Is there any benefit from insisting that the notions of distances and angles are meaningless even in such a coordinate-equipped affine space? After all, once coordinates are present one can immediately define an inner product, and with it angles, and a norm, and with the latter also a distance.

The reason I ask this question is that in a series of lectures on linear algebra that I'm watching, the lecturer uses coordinates throughout (and all spaces are finite-dimensional), while at the same time repeatedly reminding the audience that angles and distances are not defined in an affine space. This seems a bit silly to me: it's fine to say that angles and distances are meaningless in a general affine space, but once one introduces coordinates in the affine space, it seems to me one also introduces angles and distances. No?

In the above I've confined myself to the case of finite-dimensional spaces. The situation for infinite-dimensional affine spaces is not so clear-cut in my mind, because in this case, AFAICT, the existence of a basis does not imply coordinates and inner products.

However, if we add an inner product to the (linear part of the) affine space structure (i.e. considering the triple $(A,V,\langle-,-\rangle)$), then we can calmly refer to the inner product and lengths, angles.