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In mathematics, a **normed vector space** or **normed space** is a vector space over the real or complex numbers on which a norm is defined.^{[1]} A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms:

- Non-negativity: for every ,.
- Positive definiteness: for every , if and only if is the zero vector.
- Absolute homogeneity: for every and ,
- Triangle inequality: for every and ,

If is a real or complex vector space as above, and is a norm on , then the ordered pair is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by .

A norm induces a distance, called its *(norm) induced metric*, by the formula

which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a

An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance by the formula

The study of normed spaces and Banach spaces is a fundamental part of functional analysis, a major subfield of mathematics.

**^**Callier, Frank M. (1991).*Linear System Theory*. New York: Springer-Verlag. ISBN 0-387-97573-X.