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[edit]

1. The Euclidean space ${\displaystyle {\mathbb{ℝ}}^n}$ endowed with the Euclidean norm ${\displaystyle \Vert x\Vert =\sqrt{{\displaystyle \sum _{k=1}^n}|x_k{|}^2}}$ for all ${\displaystyle x\in {\mathbb{ℝ}}^n}$. 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 ${\displaystyle \Vert f\Vert ={\mathrm{e}\mathrm{s}\mathrm{s}\sup }_{x\in [0,1]}|f(x)|}$. This is an example of an infinite dimensional normed space.

See also[edit]

• Completeness
• Inner product space
• Banach space
• Hilbert space