What differences are between $\mathbb{E}^n$ and $\mathbb{R}^n$ What differences are between the two notations $\mathbb{E}^n$ and $\mathbb{R}^n$?
Do they represent/define the same  space set with the same structure(s)?
Thanks and regards!
 A: I am not sure if this is standard notation, but if an author distinguishes between $\mathbb{R}^n$ and $\mathbb{E}^n$, the former may refer to the real $n$-vector space, whereas the latter also include the structure of an inner product space.
The Wikipedia article seems to agree with this.
A: In my experience $R^n$ is a model for $E^n$. $E^n$ is the axiomatic description of Euclidean space, while $R^n$ is a particular model, i.e., it satisfies the axioms of Euclidean space.
A: Tim: I think:
PlanetMath: Euclidean vector space
should also help. I was thinking more of $R^n$ more as a choice of coordinates for $E^n$, than as a model
A: In my experience $\mathbb{E}^n$ tends to refer to Euclidean $n$-space in the context of a metric space - in particular when comparing to other metrics you could put on the same set (for example, a hyperbolic metric).  $\mathbb{R}^n$ refers to $n$-space under pretty much all other contexts - as a topological space, a vector space, a set, an abelian group, or any other situation where it's not important to distinguish between the standard Euclidean metric and other metrics on $\mathbb{R}^n$.
