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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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They're the same object. Usually $\mathbb E^n$ has the implication of interpreting it as a geometric object -- Euclidean space. $\mathbb R^n$ tends to be a little more flexible in its interpretation but sometimes one might think of it just as a vector space over $\mathbb R$, so there is a chosen point in the space called the origin -- this point you basically forget about when thinking about $\mathbb E^n$. –  Ryan Budney Apr 20 '11 at 21:30
    
Define what you mean by space. –  lhf Apr 20 '11 at 22:29
    
@lhf: I mean a set with some structure(s). –  Tim Apr 20 '11 at 22:37
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4 Answers

up vote 10 down vote accepted

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 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? –  Tim Apr 21 '11 at 5:07
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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 $\mathbb{E}^n$? –  Tim Apr 22 '11 at 23:10
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