Definition:Euclidean Norm
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Definition
Let $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be a vector in the Euclidean $n$-space $\R^n$.
The Euclidean norm of $\mathbf v$ is defined as:
- $\ds \norm {\mathbf v} = \paren {\sum_{k \mathop = 1}^n v_k^2}^{1/2}$
Also see
- Euclidean Space is Normed Vector Space, proving that the Euclidean norm is a norm.
- Definition:Euclidean Metric
Generalizations
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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Euclidean norm
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euclidean norm
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euclidean norm
- 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (previous) ... (next): $1.1$: Basic Definitions
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Euclidean norm
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): $p$-norm
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces