, a norm is a function
from a real or complex vector space
to the nonnegative 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 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 pseudonorm or seminorm satisfies the first two properties of a norm, but may be zero for other vectors 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''.
Given a vector space
over a subfield
of the complex numbers
, a norm on is a nonnegative-valued real-valued function
with the following properties, where denotes the usual absolute value
For all in and all in ,
# (being ''subadditive
'' or satisfying the ''triangle inequality
# (being ''absolutely homogeneous
'' or ''absolutely scalable'').
# If then is the zero vector
(being ''positive definite'' or being ''point-separating'').
on is a function
with the properties 1 and 2 above.
Suppose that and are two norms (or seminorms) on a vector space . Then and are called equivalent, if there exist two real constants and with such that for every vector ,
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.
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.
and related markup languages, the double bar of norm notation is entered with the macro
, which renders as
The double vertical line used to denote parallel line
s, parallel operator
and parallel addition
is entered with
and is rendered as
Although looking similar, these two macros must not be confused as
denotes a bracket
denotes an operator. Therefore, their size and the spaces around them are not computed in the same way. Similarly, the single vertical bar is coded as
when used as a bracket, and as
when used as an operator.
, the code point of the "double vertical line" character ‖ is U+2016. The "double vertical line" symbol should not be confused with the "parallel to" symbol, Unicode U+2225 ( ∥ ), which is intended to denote parallel lines and parallel operators. The double vertical line should also not be confused with Unicode U+01C1 ( ǁ ), aimed to denote lateral clicks
The single vertical line | is called "vertical line" in Unicode and its code point is U+007C.
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 0) to is a norm on . There are also a large number of norms that exhibit additional properties that make them useful for specific problems.
The absolute value
is a norm on the one-dimensional
vector spaces formed by the real
or complex number
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
, 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 nonzero vector by the inverse of its 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 vector
s over an orthonormal basis
Hence, the Euclidean norm can be written in a coordinate-free way as
The Euclidean norm is also called the ''L''2
norm, 2-norm, or square norm; see ''L''''p'' space
It defines a distance function
called the Euclidean length, ''L''2
distance, or ℓ2
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 algebra
s over the real number
s. These are the real numbers
, the complex numbers
, the quaternion
, and lastly the octonion
, where the dimensions of these spaces over the real numbers are , , , and , respectively.
The canonical norms on
are their absolute value
functions, as discussed previously.
The canonical norm on
s is defined by
for every quaternion
This is the same as the Euclidean norm on
considered as the vector space
Similarly, the canonical norm on the octonion
s is the just 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:
is represented as a column vector
('x''1; ''x''2; ...; ''x''''n''
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
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
The 1-norm is simply the sum of the absolute values of the columns.
is not a norm because it may yield negative results.
Let be a real number.
The -norm (also called
-norm) of vector
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.
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
is used for absolute value of each component of the vector.
For the special case of , this becomes
Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)
is some vector such that
The set of vectors whose infinity norm is a given constant, , forms the surface of a hypercube
with edge length 2''c''.
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
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
, 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
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.
, 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 function
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
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 (octahedron
s) and the maximum norm (prism
s 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
All the above formulas also yield norms on
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 homogenous of degree , the Galois-theoretic norm is not a norm in the sense of this article. However, the th
root of the norm (assuming that concept makes sense), is a norm.
The concept of norm
in composition algebra
s does ''not'' share the usual properties of a norm as it may be negative or zero for ''z'' ≠ 0. A composition algebra (''A'', *, ''N'') consists of an algebra over a field
''A'', an involution
*, and a quadratic form
which is called the "norm".
The characteristic feature of composition algebras is the homomorphism
property of ''N'': for the product ''wz'' of two elements ''w'' and ''z'' of the composition algebra, its norm satisfies
, and O the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form
. In other composition algebras the norm is an isotropic quadratic form
For any norm ''p'' on a vector space ''V'', the reverse triangle inequality
holds: for all u and ,
:''p''(u ± v) ≥ |''p''(u) − ''p''(v)|
If is a continuous linear map between normed space, then the norm of and the norm of the transpose
of are equal.
For the Lp
norms, we have Hölder's inequality
A special case of this is the Cauchy–Schwarz inequality
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 is a square
, for the 2-norm (Euclidean norm), it is the well-known unit circle
, while for the infinity norm, it is a different square. For any ''p''-norm, it is a superellipse
with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be convex
and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and
for a ''p''-norm).
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
is said to converge
in norm to
. Equivalently, the topology consists of all sets that can be represented as a union of open balls
. If is a normed space then for all .
Two norms ‖•‖''α''
on a vector space ''V'' are called equivalent if they induce the same topology,
which happens if and only if there exist positive real numbers ''C'' and ''D'' such that for all ''x'' in ''V''
For instance, if ''p'' > ''r'' ≥ 1 on
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
Classification of seminorms: absolutely convex absorbing sets
All seminorms on a vector space ''V'' can be classified in terms of absolutely convex absorbing subset
s ''A'' of ''V''. To each such subset corresponds a seminorm ''pA
'' called the gauge
of ''A'', defined as
''(''x'') := inf
with the property that
: ⊆ ''A'' ⊆ .
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 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
:Suppose now that (''p'') contains a single ''p'': since (''p'') is separating
, ''p'' is a norm, and is its open unit ball
. Then ''A'' is an absolutely convex bounded
neighbourhood of 0, and is continuous.
:The converse is due to Andrey Kolmogorov
: any locally convex and locally bounded topological vector space is normable
:If ''V'' is an absolutely convex bounded neighbourhood of 0, the gauge ''gV
'' (so that ) is a norm.
* Operator norm