"non-Euclidean vector spaces"

[siwelwerd]

In Remark 3.5.7, we say that the complex numbers are a 'non-Euclidean vector space'. However, they are isomorphic to the Euclidean vector space $\mathbb{R}^2$. I am not sure there is universal agreement that something can be isomorphic to a Euclidean vector space but not be a Euclidean vector space.

[StevenClontz]

Would you (or someone) say "Euclidean vector space" is a synonym for "finite-dimensional vector space"?

[siwelwerd]

They are isomorphic, so it feels a little weird to me to insist they are different.

I think Euclidean carries some connotation that the base field is the real numbers though. Wikipedia, citing Berger, defines Euclidean vector space as a finite dimensional real inner product space. So I guess yes, someone does say they are synonyms.

[StevenClontz]

In any case, the intent here is that $\mathbb C$ is a vector space that isn't identical to $\mathbb R^n$, although eventually students will learn it's isomorphic to $\mathbb R^2$.

I think the clearest solution #323 is to just drop "non-Euclidean" and let the instructor make the point however they feel is appropriate.