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Euclidean space
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(Definition)
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Euclidean $n$ space is a metric space $(E,d)$ with the property that the group of isometries is transitive and is isomorphic to an $n$ dimensional Euclidean vector space. To be more precise, we are saying
that there exists an $n$ dimensional Euclidean vector space $V$ with inner product $\langle \cdot,\cdot\rangle$ and a mapping $$ +: E\times V\to E $$ such that the following hold:
- For all $x,y\in E$ there exists a unique $u\in V$ satisfying $$ y=x+u,\quad d(x,y)^2=\langle u,u\rangle,$$
- For all $x,y\in E$ and all $u\in V$ we have $$ d(x+u,y+u)=d(x,y).$$
- For all $x\in E$ and all $u,v\in V$ we have $$ (x+u)+v=x+(u+v).$$
Putting it more succinctly: $V$ acts transitively and effectively on $E$ by isometries.
- The difference between Euclidean space and a Euclidean vector space is one of loss of structure. Euclidean space is a Euclidean vector space that has ``forgotten'' its origin.
- A 2-dimensional Euclidean space is often called a Euclidean plane.
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"Euclidean space" is owned by rmilson.
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Cross-references: origin, structure, difference, isometries, mapping, inner product, Euclidean vector space, isomorphic, transitive, group of isometries, property, metric space, Euclidean
There are 98 references to this entry.
This is version 13 of Euclidean space, born on 2004-04-08, modified 2006-01-22.
Object id is 5743, canonical name is EuclideanVectorSpace.
Accessed 17115 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) | | | 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations) |
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Pending Errata and Addenda
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