In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below.

A simple example is two dimensional Euclidean space R2 equipped with the "Euclidean norm" (see below). 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 on which a norm is defined is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a given vector space in more than one way.

Given a vector space V over a subfield F of the complex numbers, a norm on V is a nonnegative-valued scalar function p: V → [0,+∞) with the following properties:^[1]

For all a ∈ F and all u, v ∈ V,

A seminorm on V is a function p : V → R with the properties 1 and 2 above.

Every vector space V with seminorm p 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 defined by:

Two norms (or seminorms) p and q on a vector space V are equivalent if there exist two real constants c and C, with c > 0, such that

A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm). Normability of topological vector spaces is characterized by Kolmogorov's normability criterion.

If a norm p : V → R is given on a vector space V then the norm of a vector v ∈ V is usually denoted by enclosing it within double vertical lines: ‖v‖ = p(v). Such notation is also sometimes used if p is only a seminorm.

For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation |v| with single vertical lines is also widespread.

In Unicode, the code point of the "double vertical line" character ‖ is U+2016. The double vertical line should not be confused with the "parallel to" symbol, Unicode U+2225 ( ∥ ), which is used in geometry to signify parallel lines and in network theory, various fields of engineering and applied electronics as parallel addition operator. This is usually not a problem because the former is used in parenthesis-like fashion, whereas the latter is used as an infix operator. The double vertical line used here should also not be confused with the symbol used to denote lateral clicks in linguistics, Unicode U+01C1 ( ǁ ). The single vertical line | is called "vertical line" in Unicode and its code point is U+007C.

In LaTeX and related markup languages, the macros | and \parallel are often used to denote a norm.

The absolute value

is a norm on the one-dimensional vector spaces formed by the real or complex numbers.

The absolute value norm is a special case of the L1 norm.

On an n-dimensional Euclidean space Rn, the intuitive notion of length of the vector x = (x1, x2, ..., x**n) 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.^[2]

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. However, all these norms are equivalent in the sense that they all define the same topology.

On an n-dimensional complex space Cn the most common norm is

In both cases the norm can be expressed as the square root of the inner product of the vector and itself:

where x is represented as a column vector ([x1; x2; ...; 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, ℓ2 distance, L, or ℓ2 norm; see L**p space.

The set of vectors in Rn+1 whose Euclidean norm is a given positive constant forms an n-sphere.

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 of dimension equivalent to that of the norm minus 1. The Taxicab norm is also called the ******. The distance derived from this norm is called the Manhattan distance or ******.

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

In contrast,

is not a norm because it may yield negative results.

The p-norm is related to the generalized mean or power mean.

This definition is still of some interest for 0 < p < 1, but the resulting function does not define a norm,^[3] because it violates the triangle inequality. What is true for this case of 0 < p < 1, even in the measurable analog, is that the corresponding Lp 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.

The partial derivative of the p-norm is given by

The derivative with respect to x, therefore, is

For the special case of p = 2, this becomes

or

The set of vectors whose infinity norm is a given constant, c, forms the surface of a hypercube with edge length 2c.

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" with quotation marks. Following Donoho's notation, the zero "norm" of x is simply the number of non-zero coordinates 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 p-norms as p 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 omit Donoho's quotation marks and inappropriately call the number-of-nonzeros function the L0 norm, echoing the notation for the Lebesgue space of measurable functions.

Other norms on Rn can be constructed by combining the above; for example

is a norm on R4.

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 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base).

There are examples of norms that are not defined by "entrywise" formulas. For instance, the Minkowski functional of a centrally-symmetric convex body in Rn (centered at zero) defines a norm on Rn.

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

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

The generalization of the above norms to an infinite number of components leads to ℓ p**and L p spaces, with norms

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

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

In particular,

i.e.,

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.

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 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:

Thus, p(u ± v) ≥ |p(u) − p(v)|.

For the Lp norms, we have Hölder's inequality^[6]

A special case of this is the Cauchy–Schwarz inequality:^[6]

All seminorms on a vector space V can be classified in terms of absolutely convex absorbing subsets A of V. To each such subset corresponds a seminorm pA called the gauge of A, defined as

with the property that

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 family (p) of seminorms p that separates points: 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.

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

norm case:

There are several generalizations of norms and semi-norms. If p is absolute homogeneity but in place of subadditivity we require that

then p satisfies the triangle inequality but is called a quasi-seminorm and the smallest value of b for which this holds is called the p; if in addition p separates points then it is called a quasi-norm.

On the other hand, if p satisfies the triangle inequality but in place of absolute homogeneity we require that

then p is called a k.

We have the following relationship between quasi-seminorms and k-seminorms:

The concept of norm in composition algebras does not share the usual properties of a norm. A composition algebra (A, *, N) consists of an algebra over a field A, an involution *, and a quadratic form N, which is called the "norm". In several cases N is an isotropic quadratic form so that A has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.