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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!

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    @lhf: I mean a set with some structure(s).2011-04-20

4 Answers 4

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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 $\mathbb{R}^n$.

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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.

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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.

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    Thanks! (1) This reply seems consistent with http://planetmath.org/encyclopedia/EuclideanVectorSpace2.html . (2) I was wondering what the axiomatic description of Euclidean space is? Is the definition of dot product part of it? Does it require the underlying set to be an algebraic structure so that the dot product can make sense?2011-04-21
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Tim: I think:

http://planetmath.org/?op=getobj&from=objects&name=EuclideanVectorSpace s

should also help. I was thinking more of $R^n$ more as a choice of coordinates for $E^n$, than as a model

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    Thanks! I guessed you are the same gary replied earlier. (1) In the link you gave, $\mathbb{E}^n$ is defined as a metric space, I was wondering how to understand a choice of coordinates for a metric space $\mathbb{E}^n$? (2) By "a model for $\mathbb{E}^n$" here and earlier, do you mean an example of $\mathbb{E}^n$?2011-04-22