 Norm (mathematics)

This article is about linear algebra and analysis. For field theory, see Field norm. For ideals, see Norm of an ideal. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering.
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector. A seminorm, on the other hand, is allowed to assign zero length to some nonzero vectors.
A simple example is the 2dimensional Euclidean space R^{2} equipped with the Euclidean norm. Elements in this vector space (e.g., (3, 7)) are usually drawn as arrows in a 2dimensional cartesian coordinate system starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude.
A vector space with a norm is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space.
Contents
Definition
Given a vector space V over a subfield F of the complex numbers a norm on V is a function p: V → F with the following properties:^{[1]}
For all a ∈ F and all u, v ∈ V,
 p(av) = a p(v), (positive homogeneity or positive scalability).
 p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).
 If p(v) = 0 then v is the zero vector (separates points).
A simple consequence of the first two axioms, positive homogeneity and the triangle inequality, is p(0) = 0 and thus
 p(v) ≥ 0 (positivity).
A seminorm is a norm with the 3rd property (separating points) removed.
Although every vector space is seminormed (e.g., with the trivial seminorm in the Examples section below), it may not be normed. Every vector space V with seminorm p(v) induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is clearly welldefined and is given by:
 p(W + v) = p(v).
A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm).
The norm of a vector v is usually denoted v, and sometimes v. However, the latter notation is generally discouraged, because it is also used to denote the absolute value of scalars and the determinant of matrices.
Examples
 All norms are seminorms.
 The trivial seminorm, with p(x) = 0 for all x in V.
 The absolute value is a norm on the real numbers.
 Every linear form f on a vector space defines a seminorm by x → f(x).
Euclidean norm
Main article: Euclidean distanceOn an ndimensional Euclidean space R^{n}, the intuitive notion of length of the vector x = (x_{1}, x_{2}, ..., x_{n}) is captured by the formula
This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. The Euclidean norm is by far the most commonly used norm on R^{n}, 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 an ndimensional complex space C^{n} the most common norm is
In both cases we can also express the norm as the square root of the inner product of the vector and itself:
where x is represented as a column vector ([x_{1}; x_{2}; ...; x_{n}]), and x* denotes its conjugate transpose.
This formula is valid for any inner product space, including Euclidean and complex spaces. For Euclidean spaces, the inner product is equivalent to the dot product. Hence, in this specific case the formula can be also written with the following notation:
The Euclidean norm is also called the Euclidean length, L^{2} distance, ℓ^{2} distance, L^{2} norm, or ℓ^{2} norm; see L^{p} space.
The set of vectors in R^{n+1} whose Euclidean norm is a given positive constant forms an nsphere.
Euclidean norm of a complex number
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 R^{2}. This identification of the complex number x + iy as a vector in the Euclidean plane, makes the quantity (as first suggested by Euler) the Euclidean norm associated with the complex number.
Taxicab norm or Manhattan norm
Main article: Taxicab geometryThe name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x.
The set of vectors whose 1norm is a given constant forms the surface of a cross polytope of dimension equivalent to that of the norm minus 1. The Taxicab norm is also called the L_{1} norm. The distance derived from this norm is called the Manhattan distance or L_{1} distance.
The one norm is simply the maximum column sum of absolute values.
In contrast,
is not a norm because it may yield negative results.
pnorm
Main article: L^{p} spaceLet p ≥ 1 be a real number.
Note that for p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and for p = we get the infinity norm or maximum norm.
This definition is still of some interest for 0 < p < 1, but the resulting function does not define a norm,^{[2]} because it violates the triangle inequality. What is true, even in the measurable analog, is that the corresponding L^{p} class is a vector space, and it is also true that the function
(without pth root) defines a distance that makes L^{p}(X) into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory, and harmonic analysis.
However, outside trivial cases, this topological vector space is not locally convex and has no continuous nonzero linear forms. Thus the topological dual space contains only the zero functional.
Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)
Main article: Maximum normThe set of vectors whose infinity norm is a given constant, c, forms the surface of a hypercube with edge length 2c.
The infinity norm is simply the maximum row sum of absolute values.
Zero norm
In probability and functional analysis, the zero norm induces a complete metric topology for the space of measureable functions and for the Fspace of sequences with F–norm , which is discussed by Stefan Rolewicz in Metric Linear Spaces.^{[3]}
Hamming distance of a vector from zero
See also: Hamming distance and discrete metricIn metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinatewise 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 nonzero point; indeed, the distance from zero remains one as its nonzero 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 componentwise to vectors, the discrete distance from zero behaves like a nonhomogeneous "norm", which counts the number of nonzero components in its vector argument; again, this nonhomogeneous "norm" is discontinuous.
In signal processing and statistics, David Donoho referred to the zero "norm" with quotation marks. Following Donoho's notation, the zero "norm" of x is simply the number of nonzero elements of x, or the Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of pnorms as p approaches 0. Of course, the zero "norm" is not a Bnorm, because it is not positive homogeneous. It is not even an Fnorm, because it is discontinuous, jointly and severally, with respect to the scalar argument in scalarvector multiplication and with respect to its vector argument. Abusing terminology, some engineers omit Donoho's quotation marks and inappropriately call the numberofnonzeros function the L0 norm (sic.), also misusing the notation for the Lebesgue space of measurable functions.
Other norms
Other norms on R^{n} can be constructed by combining the above; for example
is a norm on R^{4}.
For any norm and any injective linear transformation A we can define a new norm of x, equal to
In 2D, with A a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. In 2D, each A 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 1norm (octahedrons) and the maximum norm (prisms with parallelogram base).
All the above formulas also yield norms on C^{n} without modification.
Infinitedimensional case
The generalization of the above norms to an infinite number of components leads to the L^{p} spaces, with norms
(for complexvalued sequences x resp. functions f defined on ), which can be further generalized (see Haar measure).
Any inner product induces in a natural way the norm
Other examples of infinite dimensional normed vector spaces can be found in the Banach space article.
Properties
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1norm the unit circle in R^{2} is a square, for the 2norm (Euclidean norm) it is the wellknown unit circle, while for the infinity norm it is a different square. For any pnorm it is a superellipse (with congruent axes). See the accompanying illustration. Note that due to the definition of the norm, the unit circle is always convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle).
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 of vectors {v_{n}} is said to converge in norm to v if as . Equivalently, the topology consists of all sets that can be represented as a union of open balls.
Two norms •_{α} and •_{β} on a vector space V are called equivalent if there exist positive real numbers C and D such that
for all x in V. For instance, on , if p > r > 0, then
In particular,
If the vector space is a finitedimensional real/complex one, all norms are equivalent. If not, some norms are not.
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.
Every (semi)norm is a sublinear function, which implies that every norm is a convex function. As a result, finding a global optimum of a normbased objective function is often tractable.
Given a finite family of seminorms p_{i} on a vector space the sum
is again a seminorm.
For any norm p on a vector space V, we have that for all u and v ∈ V:
 p(u ± v) ≥  p(u) − p(v) 
For the l^{p} norms, we have the Hölder's inequality^{[4]}
A special case of the above property is the Cauchy–Schwarz inequality:^{[4]}
Classification of seminorms: Absolutely convex absorbing sets
All seminorms on a vector space V can be classified in terms of absolutely convex absorbing sets in V. To each such set, A, corresponds a seminorm p_{A} called the gauge of A, defined as
 p_{A}(x) := inf{α : α > 0, x ∈ α A}
with the property that
 {x : p_{A}(x) < 1} ⊆ A ⊆ {x : p_{A}(x) ≤ 1}.
Conversely:
Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common method to construct such a basis is to use a separating family (p) of seminorms p: the collection of all finite intersections of sets {p<1/n} turns the space into a locally convex topological vector space so that every p is continuous.
A such method is used to design weak and weak* topologies.
norm case:
 Suppose now that (p) contains a single p: since (p) is separating, p is a norm, and A={p<1} is its open unit ball. Then A is an absolutely convex bounded neighbourhood of 0, and p = p_{A} is continuous.
 The converse is due to Kolmogorov: any locally convex and locally bounded topological vector space is normable. Precisely:
 If V is an absolutely convex bounded neighbourhood of 0, the gauge g_{V} (so that V={g_{V} <1}) is a norm.
See also
 Asymmetric norm, a generalization of a norm for which x and −x are not necessarily equal.
 Matrix norm
 Mahalanobis distance
 Manhattan distance
 Relation of norms and metrics, a translation invariant and homogeneous metric can be used to define a norm.
 Normed vector space
 Oneform
Notes
 ^ Prugovec̆ki 1981, page 20
 ^ Except in R^{1}, where it coincides with the Euclidean norm, and R^{0}, where it is trivial.
 ^ Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi+524, ISBN 9027721866, MR920371, OCLC 13064804
 ^ ^{a} ^{b} Golub, Gene; Charles F. Van Loan (1996). Matrix Computations  Third Edition. Baltimore: The Johns Hopkins University Press. p. 53. ISBN 080185413X.
References
Categories: Norms (mathematics)
 Linear algebra
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