Why $E^2$ instead of $\Bbb R^2$

Question

I was reading a paper earlier where whenever the author would discuss vectors in polar coordinates, he'd call the space $E^2$. He'd even give a function of vectors in that space as $f:E^2 \to \Bbb R$. I'm wondering why he didn't just call that space $\Bbb R^2$. Is there some difference between the spaces themselves if you decide to use polar rather than Cartesian coordinates? Is this a standard notation? Does the $E$ stand for anything (like $\Bbb R$ is short for "reals")?

Answer 1

Sometimes $E$ is used to emphasize that the space is Euclidean. This could mean, for example, that the space comes with an inner product (I have seen this used at least once). The particular meaning of the notation is not standard across all disciplines. As to whether the space is the same, topologically the answer is yes, but the notation can be used to clarify what use the space is being put to.

Answer 2

$E$ probably stands for Euclidean.

Quoting from Wikipedia:

Mathematicians denote the $n$-dimensional Euclidean space by $\mathbf E^n$ if they wish to emphasize its Euclidean nature, but $\mathbb R^n$is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished