Euclidean vector space
1 Definition
The term Euclidean vector space is synonymous with finite-dimensional, real, positive definite, inner product space. The canonical example is , equipped with the usual dot product. Indeed, every Euclidean vector space is isomorphic to , up to a choice of orthonormal basis of . As well, every Euclidean vector space carries a natural metric space structure given by
2 Remarks.
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An analogous object with complex numbers as the base field is called a unitary space.
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Dropping the assumption of finite-dimensionality we arrive at the class of real pre-Hilbert spaces.
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If we drop the inner product and the vector space structure, but retain the metric space structure, we arrive at the notion of a Euclidean space.
Title | Euclidean vector space |
Canonical name | EuclideanVectorSpace |
Date of creation | 2013-03-22 15:38:24 |
Last modified on | 2013-03-22 15:38:24 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 9 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 15A63 |
Related topic | InnerProductSpace |
Related topic | UnitarySpace |
Related topic | PositiveDefinite |
Related topic | EuclideanDistance |
Related topic | Vector |
Related topic | EuclideanVectorSpace |