# norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. A seminorm (or pseudonorm) on the other hand is allowed to assign zero length to some non-zero vectors.

A simple example is the 2-dimensional Euclidean space R2 equipped with the Euclidean norm. Elements in this vector space (e.g., (3, 7) ) are usually drawn as arrows in a 2-dimensional 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.

## Definition

Given a vector space V over a subfield F of the complex numbers such as the complex numbers themselves or the real or rational numbers, a seminorm on V is a function p:VR; xp(x) with the following properties:

For all a in F and all u and v in V,
1. p(a v) = |a| p(v), (positive homogeneity or positive scalability)
2. p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).

A simple consequence of these two axioms, positive homogeneity and the triangle inequality, is p(0) = 0 and thus
p(v) ≥ 0 (positivity).

A norm is a seminorm with the additional property
p(v) = 0 if and only if v is the zero vector (positive definiteness).

A norm is usually denoted ||v||, and sometimes |v|, instead of p(v).

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 given by ||W+v|| = p(v) and is clearly well-defined.

A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm).

## 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

On Rn, the intuitive notion of length of the vector x = [x1, x2, ..., xn] 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 Rn, but there are other norms on this vector space as will be shown below.

On Cn the most common norm is
, equivalent with the Euclidean norm on R2n.

In each case we can also express the norm as the square root of the inner product of the vector and itself. The euclidean norm is also called the l 2, see Lp space.

The set of vectors whose Euclidean norm is a given constant forms the surface of a sphere.

### Taxicab norm or Manhattan norm

Main article Taxicab geometry

The 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 1-norm is a given constant forms the surface of a cross polytope.

### p-norm

Let p≥1 be a real number.
Note that for p = 1 we get the taxicab norm and for p = 2 we get the Euclidean norm. See also Lp space.

### Infinity norm or maximum norm

Main article maximum norm

The set of vectors whose ∞-norm is a given constant forms the surface of a hypercube.

### Zero norm

In the machine learning and optimization literature, one often finds reference to the zero norm. The zero norm of x is defined as where is the p-norm defined above. If we define then we can write the zero norm as . It follows that the zero norm of x is simply the number of non-zero elements of x. Despite its name, the zero norm is not a true norm; in particular, it is not positive homogeneous. Such a norm can be defined over arbitrary fields (besides the fields of complex numbers). In the context of the information theory, it is often called the Hamming distance in the case of the 2-element GF(2) field.

### Other norms

Other norms on Rn can be constructed by combining the above; for example
is a norm on R4.

For any norm and any bijective linear transformation A we can define a new norm of x, equal to
In 2D, with A a rotation by 45Â° 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 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base).

All the above formulas also yield norms on Cn without modification.

### Infinite dimensional case

The generalization of the above norms to an infinite number of components leads to the Lp spaces, with norms
resp.
(for complex-valued 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

Illustrations of unit circles in different norms.

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 in R2 is a rhomboid, for the 2-norm (Euclidean norm) it is the well-known unit circle, while for the infinity norm it is a square. For any p-norm it is a superellipse (with congruent axes). See the accompanying illustration.

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.

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. On a finite-dimensional vector space all norms are equivalent. For instance, the , , and norms are all equivalent on :

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 norm-based objective function is often tractable.

Given a finite family of seminorms pi 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 vV:
p(u Â± v) ≥ | p(u) − p(v) |

For the lp norms, we have
[1]
A special case of the above property is the Cauchy-Schwarz inequality:
[1]

## Absolutely convex and absorbing sets

Seminorms are closely related to absolutely convex and absorbing sets. Let p be a seminorm on a vector space V, then for any scalar α the sets {x : p(x) < α} and {x : p(x) ≤ α} are absorbing and absolutely convex. In a normed vector space the set {x : p(x) ≤ 1} is called the closed unit ball.

Conversely to each absorbing and absolutely convex subset A of V corresponds a seminorm p called the gauge of A, defined as
p(x) := inf{α : α > 0, x ∈ α A}
with the property that
{x : p(x) < 1} ⊆ A ⊆ {x : p(x) ≤ 1}.

A locally convex topological vector space has a local basis consisting of absolutely convex and absorbing sets. A common method to construct such a basis is to use a family of seminorms. Typically this family is infinite, and there are enough seminorms to distinguish between elements of the vector space, creating a Hausdorff space.

## References

1. ^ Golub, Gene; Charles F. Van Loan (1996). Matrix Computations - Third Edition. Baltimore: The Johns Hopkins University Press, 53. ISBN 0-8018-5413-X.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero. It is usually written or 0 or simply 0.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
Magnitude may refer to:
• Magnitude (mathematics), a measure of the size of a mathematical object:
• A vector object has both magnitude and direction as its defining characteristics.

Rn. It turns out that the following properties of "vector length" are the crucial ones.
1. The zero vector, 0, has zero length; every other vector has a positive length.

Rn. It turns out that the following properties of "vector length" are the crucial ones.
1. The zero vector, 0, has zero length; every other vector has a positive length.

In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
In mathematics, a complex number is a number of the form

where a and b are real numbers, and i is the imaginary unit, with the property i Â² = −1.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction , where b is not zero.
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than or equal to the sum of the other two sides but greater than or equal to the difference between the two sides.

null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero. It is usually written or 0 or simply 0.
quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation.
topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space.
Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure.
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.