Normed space

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

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