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
something that satisfies the traditional Euclid's axioms, or Hilbert's axioms?
or is it defined to be an inner product space?
or is it defined to be a set on which we can somehow define the notion of "length" and "angle"?
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?