Normed space

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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

  1. The Euclidean space \mathbb{R}^n endowed with the Euclidean norm \|x\|=\sqrt{\sum_{k=1}^{n}|x_k|^2} for all x \in \mathbb{R}^n. 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\|=\mathop{{\rm ess} \sup}_{x \in [0,1]}|f(x)|. This is an example of an infinite dimensional normed space.

See also

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