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Normed space
From Citizendium, the Citizens' Compendium
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
![\|f\|=\mathop{{\rm ess} \sup}_{x \in [0,1]}|f(x)|](/images/math/9/d/1/9d1594a79f8e21f8f0e2d61d2dc898ff.png)
. This is an example of an infinite dimensional normed space.
