Norm (mathematics)

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in an Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself.

This article is about norms of normed vector spaces. For field theory, see Field norm. For ideals, see Ideal norm. For commutative algebra, see Absolute value (algebra). For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering.

A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin.^[1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.

The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".^[1] A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality " $\,\leq \,$ " in the homogeneity axiom.^[2] It can also refer to a norm that can take infinite values,^[3] or to certain functions parametrised by a directed set.^[4]

– Generalization of the concept of a norm

Asymmetric norm

– A topological vector space whose topology can be defined by a metric

F-seminorm

Gowers norm

– All infinite-dimensional, separable Banach spaces are homeomorphic

Kadec norm

– Periodicity computation method

Least-squares spectral analysis

– Statistical distance measure

Mahalanobis distance

– Property determining comparison and ordering

Magnitude (mathematics)

– Norm on a vector space of matrices

Matrix norm

– Mathematical metric in normed vector space

Minkowski distance

– Function made from a set

Minkowski functional

– Measure of the "size" of linear operators

Operator norm

– A topological vector space whose topology can be defined by a metric

Paranorm

– Mathematical space with a notion of distance

Relation of norms and metrics

– nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenous

Seminorm

– Type of function in linear algebra

Sublinear function

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Bourbaki, Nicolas

Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. . Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.

Lecture Notes in Mathematics

Kubrusly, Carlos S. (2011). The Elements of Operator Theory (Second ed.). Boston: . ISBN 978-0-8176-4998-2. OCLC 710154895.

Birkhäuser

Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. 978-1584888666. OCLC 144216834.

ISBN

; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

Schaefer, Helmut H.

(2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

Trèves, François

(2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.