Normed space

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This editable Main Article is under development and subject to a disclaimer.

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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[edit]

1. The Euclidean space ℝn endowed with the Euclidean norm ‖x‖=∑k=1n|xk|2 for all x∈ℝn. 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.
2. The space of the equivalence class of all real valued bounded Lebesgue measurable functions on the interval [0,1] with the norm ‖f‖=esssupx∈[0,1]|f(x)|. This is an example of an infinite dimensional normed space.

## See also[edit]

* Completeness
* Inner product space
* Banach space
* Hilbert space