Normed space: Difference between revisions
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In [[mathematics]], a '''normed space''' is a [[vector space]] that is endowed with a [[norm]]. A [[completeness|complete]] normed space is called a [[Banach space]]. | In [[mathematics]], a '''normed space''' is a [[vector space]] that is endowed with a [[norm]]. A [[completeness|complete]] normed space is called a [[Banach space]]. | ||
==Examples of normed spaces== | ==Examples of normed spaces== | ||
#The Euclidean space <math>\mathbb{R}^n</math> endowed with the Euclidean norm <math>\|x\|=\sqrt{\sum_{k=1}^{n}|x_k|^2}</math> for all <math>x \in \mathbb{R}^n</math>. This is the canonical example of a ''finite dimensional'' vector space; in fact ''all'' finite dimensional real normed spaces of dimension ''n'' are isomorphic to this space and, indeed, to one another. | #The Euclidean space <math>\mathbb{R}^n</math> endowed with the Euclidean norm <math>\|x\|=\sqrt{\sum_{k=1}^{n}|x_k|^2}</math> for all <math>x \in \mathbb{R}^n</math>. This is the canonical example of a ''finite dimensional'' vector space; in fact ''all'' finite dimensional real normed spaces of dimension ''n'' are isomorphic to this space and, indeed, to one another. | ||
#The space of the [[equivalence class]] of all real valued [[bounded set|bounded]] [[ | #The space of the [[equivalence class]] of all real valued [[bounded set|bounded]] [[Lebesgue measurable]] functions on the interval [0,1] with the norm <math>\|f\|=\mathop{{\rm ess} \sup}_{x \in [0,1]}|f(x)|</math>. This is an example of an ''infinite dimensional'' normed space. | ||
==See also== | ==See also== | ||
[[Completeness]] | * [[Completeness]] | ||
* [[Inner product space]] | |||
[[Inner product space]] | * [[Banach space]] | ||
* [[Hilbert space]] | |||
[[Banach space]] | |||
[[Hilbert space | |||
Revision as of 08:59, 16 November 2007
In mathematics, a normed space is a vector space that is endowed with a norm. A complete normed space is called a Banach space.
Examples of normed spaces
- The Euclidean space endowed with the Euclidean norm for all . This is the canonical example of a finite dimensional vector space; in fact all finite dimensional real normed spaces of dimension n are isomorphic to this space and, indeed, to one another.
- The space of the equivalence class of all real valued bounded Lebesgue measurable functions on the interval [0,1] with the norm . This is an example of an infinite dimensional normed space.
![{\displaystyle \|f\|=\mathop {{\rm {ess}}\sup } _{x\in [0,1]}|f(x)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e82e88f67608bab6e63568ab4259d48f9cd2e6)