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 a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself.

A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. 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". It can also refer to a norm that can take infinite values or to certain functions parametrised by a directed set.

Definition

Given a vector space over a subfield a norm is a real-valued function denotes the usual absolute value:

1. Subadditivity / Triangle inequality:

2. Absolute homogeneity:

Positive definiteness / Positiveness / :

A seminorm that has properties (1.) and (2.) so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms.

also has the following property:

1. Non-negativity:

Some authors include non-negativity as part of the definition of "norm", although this is not necessary. Although this article defined "" to be a synonym of "positive definite", some authors instead define "" to be a synonym of "non-negative"; these definitions are not equivalent.

Notation

, as proposed by Stefan Banach is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below with single vertical lines is also widespread.

Examples

is a Hamel basis There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

Absolute-value norm

The absolute value is a norm on the vector space formed by the real or complex numbers. The complex numbers form a one-dimensional vector space are real numbers, form a norm.

is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.

Euclidean norm

-dimensional Euclidean space is captured by the formula

This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem. This operation may also be referred to as "SRSS", which is an acronym for the square root of the sum of squares.

but there are other norms on this vector space as will be shown below. However, all these norms are equivalent in the sense that they all define the same topology on finite-dimensional spaces.

The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis. Hence, the Euclidean norm can be written in a coordinate-free

The Euclidean norm is also called the quadratic norm, norm, norm, 2-norm, or square norm; see L^p space. It defines a distance function called the Euclidean length, distance, or distance.

whose Euclidean norm is a given positive constant forms an n-sphere.

Euclidean norm of complex numbers

The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane is the complex conjugate

Quaternions and octonions

There are exactly four Euclidean Hurwitz algebrass over the real numbers the quaternions and lastly the octonions are their absolute value functions, as discussed previously.

of quaternions Similarly, the canonical norm on the octonions

Finite-dimensional complex normed spaces

-dimensional complex space

In this case, the norm can be expressed as the square root of the inner product is represented as a column vector denotes its conjugate transpose.

This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product

Taxicab norm or Manhattan norm

The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan

The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope, which has dimension equal to the dimension of the vector space minus 1. The Taxicab norm is also called the norm. The distance derived from this norm is called the Manhattan distance or distance.

The 1-norm is simply the sum of the absolute values of the columns.

is not a norm because it may yield negative results.

p-norm

is we get the taxicab norm we get the Euclidean norm-norm approaches the infinity norm or maximum norm-norm is related to the generalized mean or power mean.

-norm is even induced by a canonical inner product This inner product can be expressed in terms of the norm by using the polarization identity this inner product is the associated with a measure space which consists of all square-integrable functions

but the resulting function does not define a norm, because it violates the triangle inequality into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory and harmonic analysis. However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional.

denotes Hadamard product is used for absolute value of each component of the vector.

Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)

forms the surface of a hypercube

Energy norm

The energy norm is defined in terms of a symmetric positive definite as

is the identity matrix, this norm corresponds to the Euclidean norm is diagonal, this norm is also called a weighted norm. The energy norm is induced by the inner product.

In general, the value of the norm is dependent on the spectrum is bounded from below and above by the smallest and largest absolute eigenvalues coincides with the corresponding (normalized) eigenvectors. Based on the symmetric matrix square root, the energy norm of a vector can be written in terms of the standard Euclidean norm as

Zero norm

In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space Here we mean by F-norm The F-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.

Hamming distance of a vector from zero

In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory. In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.

In signal processing and statistics, David Donoho referred to the zero "norm" approaches 0. Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous. Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument. Abusing terminology, some engineers norm, echoing the notation for the Lebesgue space of measurable functions.

Infinite dimensions

The generalization of the above norms to an infinite number of components leads to \ell^pL^p spaces with norms

respectively, which can be further generalized (see Haar measure, giving a supremum norm

Any inner product

Other examples of infinite-dimensional normed vector spaces can be found in the Banach space article.

space gives a strictly finer topology

Composite norms

For any norm and any injective linear transformation applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a parallelogram of a particular shape, size, and orientation.

In 3D, this is similar but different for the 1-norm (octahedrons) and the maximum norm (prismss with parallelogram base).

There are examples of norms that are not defined by "entrywise" formulas. For instance, the Minkowski functional (see below).

without modification.

There are also norms on spaces of matrices (with real or complex entries), the so-called matrix norms.

In abstract algebra

be a finite extension of inseparable degree If the distinct embeddings then the Galois-theoretic norm As that function is homogeneous of degree [E : k]-th root of the norm (assuming that concept makes sense) is a norm.

Composition algebras

in composition algebras does share the usual properties of a norm since null vectors consists of an algebra over a field an involution and a quadratic form N(z) = z z^* called the "norm".

The characteristic feature of composition algebras is the homomorphism In the case of division algebras the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In the split algebras the norm is an isotropic quadratic form.

Properties

the reverse triangle inequality and the norm of the transpose are equal.

For the L^p norms, we have Hölder's inequality A special case of this is the Cauchy–Schwarz inequality:

Every norm is a seminorm and thus satisfies all properties of the latter. In turn, every seminorm is a sublinear function and thus satisfies all properties of the latter. In particular, every norm is a convex function.

Equivalence

The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle-norm, it is a superellipse with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be convex-norm).

In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence is said to converge Equivalently, the topology consists of all sets that can be represented as a union of open balls is a normed space then

are called if they induce the same topology, then

If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.

Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise, the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic. This form of equivalence should not be mistaken to imply that norms are always interchangeable. In the context of model fitting, for example, different norms can lead to different fits and different algorithm efficiencies.

Classification of seminorms: absolutely convex absorbing sets

can be classified in terms of absolutely convex absorbing subsets called the gauge is the infimum Conversely:

Any locally convex topological vector space has a local basis that separates points turns the space into a locally convex topological vector space so that every p is continuous.

Such a method is used to design weak and weak* topologies.

norm case:

1. is separating is its open unit ball is an absolutely convex bounded is continuous.

   The converse is due to Andrey Kolmogorov: any locally convex and locally bounded topological vector space is normable. Precisely:

2. is a norm.

See also

Notes

References

Bibliography