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 zero is 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 of the complex numbers a norm on is a real-valued function with the following properties, where denotes the usual absolute value of a scalar :

Subadditivity / Triangle inequality:

for all

Absolute homogeneity:

for all and all scalars

Positive definiteness / Positiveness / :

for all if then

* Because property implies some authors replace property with the equivalent condition: for every if and only if

A seminorm on is a function that has properties and so that in particular, every norm is also a seminorm. However, there exist seminorms that are not norms. Properties and imply that if is a norm, then and that also has the following property:

Non-negativity: for all

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.

Equivalent norms

Two norms and on a vector space are equivalent if there exist positive real constants and such that for every vector
The relation " is equivalent to " is reflexive, symmetric, and transitive, and thus defines an equivalence relation on the set of all norms on
The norms and are equivalent if and only if they induce the same topology on Any two norms on a finite-dimensional space are equivalent, but this does not extend to infinite-dimensional spaces.

Notation

If a norm is given on a vector space then the norm of a vector is usually denoted by enclosing it within double vertical lines:, as proposed by Stefan Banach in his doctoral thesis from 1920. Such notation is also sometimes used if is only a seminorm. For the length of a vector in Euclidean space, the notation with single vertical lines is also widespread.

Examples

Every vector space admits a norm: If is a Hamel basis for a vector space then the real-valued map that sends to is a norm on 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 over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.
Any norm on a one-dimensional vector space is equivalent to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces where is either or and norm-preserving means that
This isomorphism is given by sending to a vector of norm which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.

Euclidean norm

On the -dimensional Euclidean space the intuitive notion of length of the vector 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.
The Euclidean norm is by far the most commonly used norm on 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 way as
The Euclidean norm is also called the quadratic norm, norm, norm, 2-norm, or square norm; see space.
It defines a distance function called the Euclidean length, distance, or distance.
The set of vectors in whose Euclidean norm is a given positive constant forms an -sphere.

Euclidean norm of complex numbers

The Euclidean norm of a complex number is the absolute value of it, if the complex plane is identified with the Euclidean plane This identification of the complex number as a vector in the Euclidean plane, makes the quantity the Euclidean norm associated with the complex number. For, the norm can also be written as where is the complex conjugate of

Quaternions and octonions

There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers the complex numbers the quaternions and lastly the octonions where the dimensions of these spaces over the real numbers are respectively.
The canonical norms on and are their absolute value functions, as discussed previously.
The canonical norm on of quaternions is defined by
for every quaternion in This is the same as the Euclidean norm on considered as the vector space Similarly, the canonical norm on the octonions is just the Euclidean norm on

Finite-dimensional complex normed spaces

On an -dimensional complex space the most common norm is
In this case, the norm can be expressed as the square root of the inner product of the vector and itself:
where is represented as a column vector and 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. Hence the formula in this case can also be written using the following notation:

Taxicab norm or Manhattan norm

The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point
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.
In contrast,
is not a norm because it may yield negative results.

''p''-norm

Let be a real number.
The -norm of vector is
For we get the [|taxicab norm], for we get the Euclidean norm, and as approaches the -norm approaches the infinity norm or maximum norm:
The -norm is related to the generalized mean or power mean.
For the -norm is even induced by a canonical inner product meaning that for all vectors This inner product can be expressed in terms of the norm by using the polarization identity.
On this inner product is the defined by
while for the space associated with a measure space which consists of all square-integrable functions, this inner product is
This definition is still of some interest for but the resulting function does not define a norm, because it violates the triangle inequality.
What is true for this case of even in the measurable analog, is that the corresponding class is a vector space, and it is also true that the function
defines a distance that makes 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.
The partial derivative of the -norm is given by
The derivative with respect to therefore, is
where denotes Hadamard product and is used for absolute value of each component of the vector.
For the special case of this becomes
or

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

If is some vector such that then:
The set of vectors whose infinity norm is a given constant, forms the surface of a hypercube with edge length

Energy norm

The energy norm of a vector is defined in terms of a symmetric positive definite matrix as
It is clear that if is the identity matrix, this norm corresponds to the Euclidean norm. If is diagonal, this norm is also called a weighted norm. The energy norm is induced by the inner product given by for.
In general, the value of the norm is dependent on the spectrum of : For a vector with a Euclidean norm of one, the value of is bounded from below and above by the smallest and largest absolute eigenvalues of respectively, where the bounds are achieved if coincides with the corresponding 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 of sequences with F–norm
Here we mean by F-norm some real-valued function on an F-space with distance such that The F-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.