What really is the modern definition of Euclidean spaces?

What is the modern definition of Euclidean spaces?

I read the Wikipedia article about the topic, but I still don't get it.

Is a Euclidean space

1. something that satisfies the traditional Euclid's axioms, or Hilbert's axioms?

2. or is it defined to be an inner product space?

3. or is it defined to be a set on which we can somehow define the notion of "length" and "angle"?

4. or is it defined to be an affine space?

If a Euclidean space is defined as in case 2 (i.e. as an inner product space), then do we still need Euclid's axioms or Hilbert's axioms?

For example, there's a Hilbert's axiom

For every two points A, B there exists a line a that contains each of the points A, B.

but in terms of inner product space terminology, it can be directly proved by trivial pre-calculus technique, right?

• I am used to the following terminology : an euclidean vector space is defined as a finite dimensional real vector space, equipped with a scalar product (and hence with notions of norm, distance and (non-oriented) angle). Same object but without any condition about dimension is called a real-prehilbertian vector space. Finally, a real Hilbert space is a real prehilbertian space which also a Banach space (complete for the metric induced by the euclidean norm). Jan 15, 2017 at 14:02
• 2 is the right answer. Jan 15, 2017 at 14:02
• So an inner product space will satisfy Euclid's or Hilbert's axioms? And we can say that, since we adapt the inner product language for discussing all the geometry matters, nowadays we don't need Euclid's or Hilbert's axioms anymore right?
– Eric
Jan 15, 2017 at 14:08
• 5. There is no completely universal, standard definition of this term and either a) you're free to use your own definition (within reason), or b) you're free to use whichever definition you prefer (if you have a choice), or c) you're stuck with whatever definition the author of the book you're reading/the person lecturing you has decided is convenient for them. Jan 16, 2017 at 17:22

1. This is not widely used nowadays.

2. More precisely it would be a finite-dimensional real inner-product space.

3. This is not specific enough. This is the notion of a Riemannian manifold and includes Euclidean spaces as special cases. The sphere in $\mathbb{R}^3$ is an example of a Riemannian manifold which is not a Euclidean space.

4. A Euclidean space is in particular an affine space and every affine space can be given the structure of a Euclidean space (by choosing an origin and an inner-product). But an affine space does not come with this structure.

Now it depends on the context and in which field of mathematics you are working. Many people view it as in 2 but others have the point of view that Euclidean spaces do not have a distinguished origin. In that case it would be closer to 3 and 4, or more precisely, we would say that it is an affine space with a flat Riemannian metric. Or similarly, a common terminology in differential geometry is:

1. $\mathbb{R}^n$ with the flat metric.
• By the way, what is the relation between a model of Hilbert's geometric axioms and the 1.~4. here? Is the entities of 2. or 4. a subset of the set of all the models of Hilbert's axiom?
– Eric
Aug 22, 2017 at 15:18