In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
: it
commutes with scaling, obeys a form of the
triangle inequality, and is zero only at the origin. In particular, the
Euclidean distance of a vector from the origin is a norm, called the
Euclidean norm, or
2-norm, which may also 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".
A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "
" in the homogeneity axiom.
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/: for all
if
then
#* Because property (2.) implies
some authors replace property (3.) with the equivalent condition: for every
if and only if
A
seminorm on
is a function
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. Properties (1.) and (2.) imply that if
is a norm (or more generally, a seminorm) 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.
Equivalent norms
Suppose that
and
are two norms (or seminorms) on a vector space
Then
and
are called equivalent, if there exist two positive real constants
and
with
such that for every vector
The relation "
is equivalent to
" is
reflexive,
symmetric (
implies
), 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:
Such notation is also sometimes used if
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
with single vertical lines is also widespread.
Examples
Every (real or complex) vector space admits a norm: If
is a
Hamel basis for a vector space
then the real-valued map that sends
(where all but finitely many of the scalars
are
) 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
one-dimensional vector spaces formed by the
real or
complex numbers.
Any norm
on a one-dimensional vector space
is equivalent (up to scaling) 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.
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
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 (also called the modulus) 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
(as first suggested by Euler) the Euclidean norm associated with the complex number.
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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebr ...
. 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 (like that of the
New York borough of
Manhattan) 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 of dimension equivalent to that of the norm 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 (also called
-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 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
(without
th root) 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
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.
Hamming distance of a vector from zero
In
metric geometry, the
discrete metric
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
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
is simply the number of non-zero coordinates of
or the Hamming distance of the vector from zero.
When this "norm" is localized to a bounded set, it is the limit of
-norms as
approaches 0.
Of course, the zero "norm" is not truly a norm, because it is not
positive homogeneous
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''de ...
.
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-non-zeros function the
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
and spaces, with norms
for complex-valued sequences and functions on
respectively, 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.
Composite norms
Other norms on
can be constructed by combining the above; for example
is a norm on
For any norm and any
injective linear transformation we can define a new norm of
equal to
In 2D, with
a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each
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
(centered at zero) defines a norm on
(see below).
All the above formulas also yield norms on
without modification.
There are also norms on spaces of matrices (with real or complex entries), the so-called
matrix norms.
In abstract algebra
Let
be a
finite extension of a field
of
inseparable degree and let
have algebraic closure
If the distinct
embeddings of
are
then the Galois-theoretic norm of an element
is the value
As that function is homogeneous of degree
, the Galois-theoretic norm is not a norm in the sense of this article. However, the