Definition:Euclidean Norm

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Definition

Let $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be a vector in the real Euclidean $n$-space $\R^n$.


The Euclidean norm of $\mathbf v$ is defined as:

\(\ds \norm {\mathbf v}\) \(:=\) \(\ds \sqrt {\mathbf v \cdot \mathbf v}\) where $\mathbf v \cdot \mathbf v$ denotes the dot product
\(\ds \) \(=\) \(\ds \paren {\sum_{k \mathop = 1}^n {v_k}^2}^{1/2}\)


Also see

  • Results about Euclidean norms can be found here.


Source of Name

This entry was named for Euclid.


Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean space, Euclidean topology and Euclidean norm.

They bear that name because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.


Sources

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