View source for Norm (mathematics)

{{Short description|Length in a vector space}}
{{About|the concept in normed spaces||Norm (disambiguation)#In mathematics}}

In [[mathematics]], a '''norm''' is a [[function (mathematics)|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 (mathematics)|origin]]: it [[Equivariant map|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|Euclidean norm]], the [[#p-norm|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.<ref name="Knapp">{{cite book|title=Basic Real Analysis|publisher=Birkhäuser|author=Knapp, A.W.|year=2005|page=[https://books.google.com/books?id=4ZZCAAAAQBAJ&pg=279] |isbn=978-0-817-63250-2}}</ref> 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".<ref name="Knapp">{{cite book|title=Basic Real Analysis|publisher=Birkhäuser|author=Knapp, A.W.|year=2005|page=[https://books.google.com/books?id=4ZZCAAAAQBAJ&pg=279] |isbn=978-0-817-63250-2}}</ref> It can also refer to a norm that can take infinite values<ref>{{Cite web |title=Pseudonorm |url=https://www.spektrum.de/lexikon/mathematik/pseudonorm/8161 |access-date=2022-05-12 |website=www.spektrum.de |language=de}}</ref> or to certain functions parametrised by a [[directed set]].<ref>{{Cite journal |last=Hyers |first=D. H. |date=1939-09-01 |title=Pseudo-normed linear spaces and Abelian groups |url=http://dx.doi.org/10.1215/s0012-7094-39-00551-x |journal=Duke Mathematical Journal |volume=5 |issue=3 |doi=10.1215/s0012-7094-39-00551-x |issn=0012-7094}}</ref>

{{TOCLimit}}

==Definition==

Given a [[vector space]] <math>X</math> over a [[Field extension|subfield]] <math>F</math> of the  complex numbers <math>\Complex,</math> a '''norm''' on <math>X</math> is a [[real-valued function]] <math>p : X \to \Reals</math> with the following properties, where <math>|s|</math> denotes the usual [[absolute value]] of a scalar <math>s</math>:<ref>{{cite book|title=Real Mathematical Analysis |publisher=Springer |author=Pugh, C.C.|year=2015|page=[https://books.google.com/books?id=2NVJCgAAQBAJ&pg=PA28 page 28]|isbn=978-3-319-17770-0}} {{cite book|title=Quantum Mechanics in Hilbert Space|author=Prugovečki, E.|year=1981|page=[https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA20 page 20]}}</ref>

# [[Subadditive function|Subadditivity]]/[[Triangle inequality]]: <math>p(x + y) \leq p(x) + p(y)</math> for all <math>x, y \in X.</math>
# [[Homogeneous function|Absolute homogeneity]]: <math>p(s x) = |s| p(x)</math> for all <math>x \in X</math> and all scalars <math>s.</math>
# [[Positive definiteness]]/Positiveness{{sfn|Kubrusly|2011|p=200}}/{{Visible anchor|Point-separating}}: for all <math>x \in X,</math> if <math>p(x) = 0</math> then <math>x = 0.</math>
#* Because property (2.) implies <math>p(0) = 0,</math> some authors replace property (3.) with the equivalent condition: for every <math>x \in X,</math> <math>p(x) = 0</math> if and only if <math>x = 0.</math>

A [[seminorm]] on <math>X</math> is a function <math>p : X \to \Reals</math> that has properties (1.) and (2.)<ref>{{cite book|title=Functional Analysis|author=Rudin, W.|year=1991|page=25}}</ref> so that in particular, every norm is also a seminorm (and thus also a [[sublinear function]]al). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if <math>p</math> is a norm (or more generally, a seminorm) then <math>p(0) = 0</math> and that <math>p</math> also has the following property:

#<li value="4">[[Nonnegative|Non-negativity]]:{{sfn|Kubrusly|2011|p=200}} <math>p(x) \geq 0</math> for all <math>x \in X.</math></li>

Some authors include non-negativity as part of the definition of "norm", although this is not necessary.
Although this article defined "{{em|positive}}" to be a synonym of "positive definite", some authors instead define "{{em|positive}}" to be a synonym of "non-negative";{{sfn|Narici|Beckenstein|2011|pp=120-121}} these definitions are not equivalent.

===Equivalent norms===

Suppose that <math>p</math> and <math>q</math> are two norms (or seminorms) on a vector space <math>X.</math> Then <math>p</math> and <math>q</math> are called '''equivalent''', if there exist two positive real constants <math>c</math> and <math>C</math> such that for every vector <math>x \in X,</math>
<math display="block">c q(x) \leq p(x) \leq C q(x).</math>
The relation "<math>p</math> is equivalent to <math>q</math>" is [[Reflexive relation|reflexive]], [[Symmetric relation|symmetric]] (<math>c q \leq p \leq C q</math> implies <math>\tfrac{1}{C} p \leq q \leq \tfrac{1}{c} p</math>), and [[Transitive relation|transitive]] and thus defines an [[equivalence relation]] on the set of all norms on <math>X.</math>
The norms <math>p</math> and <math>q</math> are equivalent if and only if they induce the same topology on <math>X.</math><ref name="Conrad Equiv norms">{{cite web |url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/equivnorms.pdf |title=Equivalence of norms |last=Conrad |first=Keith |website=kconrad.math.uconn.edu |access-date=September 7, 2020 }}</ref> Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.<ref name="Conrad Equiv norms"/>

===Notation===

If a norm <math>p : X \to \R</math> is given on a vector space <math>X,</math> then the norm of a vector <math>z \in X</math> is usually denoted by enclosing it within double vertical lines: <math>\|z\| = p(z)</math>, as proposed by [[Stefan Banach]] in his doctoral thesis from 1920.  Such notation is also sometimes used if <math>p</math> is only a seminorm.  For the length of a vector in Euclidean space (which is an example of a norm, as [[#Euclidean norm|explained below]]), the notation <math>|x|</math> with single vertical lines is also widespread.

==Examples==

Every (real or complex) vector space admits a norm: If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a [[Hamel basis]] for a vector space <math>X</math> then the real-valued map that sends <math>x = \sum_{i \in I} s_i x_i \in X</math> (where all but finitely many of the scalars <math>s_i</math> are <math>0</math>) to <math>\sum_{i \in I} \left|s_i\right|</math> is a norm on <math>X.</math>{{sfn|Wilansky|2013|pp=20-21}}  There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

===Absolute-value norm===
{{redirect|Absolute-value norm|the commutative algebra concept|Absolute value (algebra)}}

The [[absolute value]]
<math>|x|</math>
is a norm on the  vector space formed by the [[real number|real]] or [[complex number]]s. The complex numbers form a [[dimension (vector space)|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 <math>p</math> on a one-dimensional vector space <math>X</math> is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving [[isomorphism]] of vector spaces <math>f : \mathbb{F} \to X,</math> where <math>\mathbb{F}</math> is either <math>\R</math> or <math>\Complex,</math> and norm-preserving means that <math>|x| = p(f(x)).</math>
This isomorphism is given by sending <math>1 \isin \mathbb{F}</math> to a vector of norm <math>1,</math> which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.

===Euclidean norm===
<!-- [[L2 norm]] and [[L2 distance]] redirect here -->
{{Further|Euclidean norm|Euclidean distance}}

On the <math>n</math>-dimensional [[Euclidean space]] <math>\R^n,</math> the intuitive notion of length of the vector <math>\boldsymbol{x} = \left(x_1, x_2, \ldots, x_n\right)</math> is captured by the formula<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Vector Norm|url=https://mathworld.wolfram.com/VectorNorm.html|access-date=2020-08-24|website=mathworld.wolfram.com|language=en}}</ref>
<math display=block>\|\boldsymbol{x}\|_2 := \sqrt{x_1^2 + \cdots + x_n^2}.</math>

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 '''s'''quare '''r'''oot of the '''s'''um of '''s'''quares.<ref>{{Cite book|title=Dynamics of Structures, 4th Ed.|last=Chopra|first=Anil|publisher=Prentice-Hall|year=2012|isbn=978-0-13-285803-8}}</ref>

The Euclidean norm is by far the most commonly used norm on <math>\R^n,</math><ref name=":1" /> 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 vector]]s over an [[orthonormal basis]].
Hence, the Euclidean norm can be written in a coordinate-free way as
<math display="block">\|\boldsymbol{x}\| := \sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}.</math>

The Euclidean norm is also called the '''quadratic norm''', '''<math>L^2</math> norm''',<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Norm|url=https://mathworld.wolfram.com/Norm.html|access-date=2020-08-24|website=mathworld.wolfram.com|language=en}}</ref> '''<math>\ell^2</math> norm''', '''2-norm''', or '''square norm'''; see [[Lp space|<math>L^p</math> space]].
It defines a [[distance function]] called the '''Euclidean length''', '''<math>L^2</math> distance''', or '''<math>\ell^2</math> distance'''.

The set of vectors in <math>\R^{n+1}</math> whose Euclidean norm is a given positive constant forms an [[n-sphere|<math>n</math>-sphere]].

====Euclidean norm of complex numbers====
{{See also|Dot product#Complex vectors}}

The Euclidean norm of a [[complex number]] is the [[Absolute value#Complex numbers|absolute value]] (also called the '''modulus''') of it, if the [[complex plane]] is identified with the [[Euclidean plane]] <math>\R^2.</math> This identification of the complex number <math>x + i y</math> as a vector in the Euclidean plane, makes the quantity <math display=inline>\sqrt{x^2 + y^2}</math> (as first suggested by Euler) the Euclidean norm associated with the complex number.  For <math>z = x +iy</math>, the norm can also be written as <math>\sqrt{\bar z z}</math> where <math>\bar z</math> is the [[complex conjugate]] of <math>z\,.</math>

===Quaternions and octonions===
{{See also|Quaternion|Octonion}}

There are exactly four [[Hurwitz's theorem (composition algebras)|Euclidean Hurwitz algebra]]s over the [[real number]]s. These are the real numbers <math>\R,</math> the complex numbers <math>\Complex,</math> the [[quaternion]]s <math>\mathbb{H},</math> and lastly the [[octonion]]s <math>\mathbb{O},</math> where the dimensions of these spaces over the real numbers are <math>1, 2, 4, \text{ and } 8,</math> respectively.
The canonical norms on <math>\R</math> and <math>\Complex</math> are their [[absolute value]] functions, as discussed previously.

The canonical norm on <math>\mathbb{H}</math> of [[quaternion]]s is defined by
<math display=block>\lVert q \rVert = \sqrt{\,qq^*~} = \sqrt{\,q^*q~} = \sqrt{\, a^2 + b^2 + c^2 + d^2 ~}</math>
for every quaternion <math>q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k</math> in <math>\mathbb{H}.</math> This is the same as the Euclidean norm on <math>\mathbb{H}</math> considered as the vector space <math>\R^4.</math> Similarly, the canonical norm on the [[octonion]]s is just the Euclidean norm on <math>\R^8.</math>

===Finite-dimensional complex normed spaces===

On an <math>n</math>-dimensional [[Complex coordinate space|complex space]] <math>\Complex^n,</math> the most common norm is
<math display=block>\|\boldsymbol{z}\| := \sqrt{\left|z_1\right|^2 + \cdots + \left|z_n\right|^2} = \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n}.</math>

In this case, the norm can be expressed as the [[square root]] of the [[inner product]] of the vector and itself:
<math display=block>\|\boldsymbol{x}\| := \sqrt{\boldsymbol{x}^H ~ \boldsymbol{x}},</math>
where <math>\boldsymbol{x}</math> is represented as a [[column vector]] <math>\begin{bmatrix} x_1 \; x_2 \; \dots \; x_n \end{bmatrix}^{\rm T}</math> and <math>\boldsymbol{x}^H</math> 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:
<math display=block>\|\boldsymbol{x}\| := \sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}.</math>

===Taxicab norm or Manhattan norm===
{{Main|Taxicab geometry}}

<math display="block">\|\boldsymbol{x}\|_1 := \sum_{i=1}^n \left|x_i\right|.</math>
The name relates to the distance a taxi has to drive in a rectangular [[street grid]] (like that of the [[New York City|New York]] borough of [[Manhattan]]) to get from the origin to the point <math>x.</math>

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 '''<math>\ell^1</math> norm'''. The distance derived from this norm is called the [[Manhattan distance]] or '''<math>\ell^1</math> distance'''.

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

In contrast,
<math display="block">\sum_{i=1}^n x_i</math>
is not a norm because it may yield negative results.

===''p''-norm===
{{Main|Lp space|l1=L<sup>p</sup> space}}

Let <math>p \geq 1</math> be a real number.
The <math>p</math>-norm (also called <math>\ell^p</math>-norm) of vector <math>\mathbf{x} = (x_1, \ldots, x_n)</math> is<ref name=":1" />
<math display="block">\|\mathbf{x}\|_p := \left(\sum_{i=1}^n \left|x_i\right|^p\right)^{1/p}.</math>
For <math>p = 1,</math> we get the [[#Taxicab norm or Manhattan norm|taxicab norm]], for <math>p = 2</math> we get the [[#Euclidean norm|Euclidean norm]], and as <math>p</math> approaches <math>\infty</math> the <math>p</math>-norm approaches the [[uniform norm|infinity norm]] or [[#Maximum_norm_.28special_case_of:_infinity_norm.2C_uniform_norm.2C_or_supremum_norm.29|maximum norm]]:
<math display="block">\|\mathbf{x}\|_\infty := \max_i \left|x_i\right|.</math>
The <math>p</math>-norm is related to the [[generalized mean]] or power mean.

For <math>p = 2,</math> the <math>\|\,\cdot\,\|_2</math>-norm is even induced by a canonical [[inner product]] <math>\langle \,\cdot,\,\cdot\rangle,</math> meaning that <math display="inline">\|\mathbf{x}\|_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}</math> for all vectors <math>\mathbf{x}.</math> This inner product can be expressed in terms of the norm by using the [[polarization identity]].
On <math>\ell^2,</math> this inner product is the ''{{visible anchor|Euclidean inner product}}'' defined by
<math display=block>\langle \left(x_n\right)_{n}, \left(y_n\right)_{n} \rangle_{\ell^2} ~=~ \sum_n \overline{x_n} y_n</math>
while for the space <math>L^2(X, \mu)</math> associated with a [[measure (mathematics)|measure space]] <math>(X, \Sigma, \mu),</math> which consists of all [[square-integrable function]]s, this inner product is
<math display=block>\langle f, g \rangle_{L^2} = \int_X \overline{f(x)} g(x)\, \mathrm dx.</math>

This definition is still of some interest for <math>0 < p < 1,</math> but the resulting function does not define a norm,<ref>Except in <math>\R^1,</math> where it coincides with the Euclidean norm, and <math>\R^0,</math> where it is trivial.</ref> because it violates the [[triangle inequality]].
What is true for this case of <math>0 < p < 1,</math> even in the measurable analog, is that the corresponding <math>L^p</math> class is a vector space, and it is also true that the function
<math display="block">\int_X |f(x) - g(x)|^p ~ \mathrm d \mu</math>
(without <math>p</math>th root) defines a distance that makes <math>L^p(X)</math> 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 <math>p</math>-norm is given by
<math display="block">\frac{\partial}{\partial x_k} \|\mathbf{x}\|_p = \frac{x_k \left|x_k\right|^{p-2}} { \|\mathbf{x}\|_p^{p-1}}.</math>

The derivative with respect to <math>x,</math> therefore, is
<math display="block">\frac{\partial \|\mathbf{x}\|_p}{\partial \mathbf{x}} =\frac{\mathbf{x} \circ |\mathbf{x}|^{p-2}} {\|\mathbf{x}\|^{p-1}_p}.</math>
where <math>\circ</math> denotes [[Hadamard product (matrices)|Hadamard product]] and <math>|\cdot|</math> is used for absolute value of each component of the vector.

For the special case of <math>p = 2,</math> this becomes
<math display="block">\frac{\partial}{\partial x_k} \|\mathbf{x}\|_2 = \frac{x_k}{\|\mathbf{x}\|_2},</math>
or
<math display="block">\frac{\partial}{\partial \mathbf{x}} \|\mathbf{x}\|_2 = \frac{\mathbf{x}}{ \|\mathbf{x}\|_2}.</math>

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

[[File:Vector norm sup.svg|frame|right|<math>\|x\|_\infty = 1</math>]]
{{Main|Maximum norm}}
If <math>\mathbf{x}</math> is some vector such that <math>\mathbf{x} = (x_1, x_2, \ldots ,x_n),</math> then:
<math display="block">\|\mathbf{x}\|_\infty := \max \left(\left|x_1\right| , \ldots , \left|x_n\right|\right).</math>

The set of vectors whose infinity norm is a given constant, <math>c,</math> forms the surface of a [[hypercube]] with edge length <math>2 c.</math>

===Energy norm===

The energy norm<ref name="SaadLinearAlgebra">{{Citation |title=Iterative Methods for Sparse Linear Systems |last=Saad |first=Yousef |year=2003 |isbn=978-0-89871-534-7 |pages=32}}</ref> of a vector <math>\boldsymbol{x} = \left(x_1, x_2, \ldots, x_n\right) \in \R^{n}</math> is defined in terms of a [[Symmetric_matrix|symmetric]] [[Definite_matrix|positive definite]] matrix <math>A \in \R^n</math> as

<math display="block">{\|\boldsymbol{x}\|}_{A} := \sqrt{\boldsymbol{x}^{T} \cdot A \cdot \boldsymbol{x}}.</math>

It is clear that if <math>A</math> is the [[identity matrix]], this norm corresponds to the [[#Euclidean_norm|Euclidean norm]]. If <math>A</math> is diagonal, this norm is also called a ''weighted norm''. The energy norm is induced by the [[Inner_product_space| inner product]] given by <math>\langle \boldsymbol{x}, \boldsymbol{y} \rangle_A := \boldsymbol{x}^{T} \cdot A \cdot \boldsymbol{y} </math> for <math>\boldsymbol{x}, \boldsymbol{y} \in \R^{n}</math>.

In general, the value of the norm is dependent on the [[Spectrum_of_a_matrix|spectrum]] of <math>A</math>: For a vector <math>\boldsymbol{x}</math> with a Euclidean norm of one, the value of <math>{\|\boldsymbol{x}\|}_{A}</math> is bounded from below and above by the smallest and largest absolute [[Eigenvalues_and_eigenvectors|eigenvalues]] of <math>A</math> respectively, where the bounds are achieved if <math>\boldsymbol{x}</math> coincides with the corresponding (normalized) eigenvectors. Based on the symmetric [[Square_root_of_a_matrix|matrix square root]] <math>A^{1/2}</math>, the energy norm of a vector can be written in terms of the standard Euclidean norm as

<math display="block">{\|\boldsymbol{x}\|}_{A} = {\|A^{1/2} \boldsymbol{x}\|}_{2}.</math>

===Zero norm===

In probability and functional analysis, the zero norm induces a complete metric topology for the space of [[measurable function]]s and for the [[F-space]] of sequences with F–norm <math display="inline">(x_n) \mapsto \sum_n{2^{-n} x_n/(1+x_n)}.</math><ref name="RolewiczControl">{{Citation |title=Functional analysis and control theory: Linear systems |last=Rolewicz |first=Stefan |year=1987 |isbn=90-277-2186-6 |publisher=D. Reidel Publishing Co.; PWN—Polish Scientific Publishers |oclc=13064804 |edition=Translated from the Polish by Ewa Bednarczuk |series=Mathematics and its Applications (East European Series) |location=Dordrecht; Warsaw |volume=29 |pages=xvi,524 |mr=920371 |doi=10.1007/978-94-015-7758-8}}</ref>
Here we mean by ''F-norm'' some real-valued function <math>\lVert \cdot \rVert</math> on an F-space with distance <math>d,</math> such that <math>\lVert x \rVert = d(x,0).</math> 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====
{{See also|Hamming distance|discrete metric}}

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 theory|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 <math>x</math> is simply the number of non-zero coordinates of <math>x,</math> or the Hamming distance of the vector from zero.
When this "norm" is localized to a bounded set, it is the limit of <math>p</math>-norms as <math>p</math> approaches 0.
Of course, the zero "norm" is '''not''' truly a norm, because it is not [[homogeneous function#Positive homogeneity|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.
[[Abuse of terminology|Abusing terminology]], some engineers{{Who|date=November 2015}} omit Donoho's quotation marks and inappropriately call the number-of-non-zeros function the <math>L^0</math> norm, echoing the notation for the [[Lp space|Lebesgue space]] of [[measurable function]]s.

===Infinite dimensions===

The generalization of the above norms to an infinite number of components leads to [[Lp space|<math>\ell^p</math> and <math>L^p</math> spaces]] for <math>p \ge 1\,,</math> with norms

<!-- The first set of \bigg is there because the sum subscript triggers a set of parenthesis that is too big, the second set is there for symmetry-->
<math display="block">\|x\|_p = \bigg(\sum_{i \in \N} \left|x_i\right|^p\bigg)^{1/p} \text{ and  }\ \|f\|_{p,X} = \bigg(\int_X |f(x)|^p ~ \mathrm d x\bigg)^{1/p}</math>

for complex-valued sequences and functions on <math>X \sube \R^n</math> respectively, which can be further generalized (see [[Haar measure]]).  These norms are also valid in the limit as <math>p \rightarrow +\infty</math>, giving a [[supremum norm]], and are called <math>\ell^\infty</math> and <math>L^\infty\,.</math>

Any [[inner product]] induces in a natural way the norm <math display=inline>\|x\| := \sqrt{\langle x , x\rangle}.</math>

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

Generally, these norms do not give the same topologies.  For example, an infinite-dimensional <math>\ell^p</math> space gives a [[finer topology|strictly finer topology]] than an infinite-dimensional <math>\ell^q</math> space when <math>p < q\,.</math>

===Composite norms===

Other norms on <math>\R^n</math> can be constructed by combining the above; for example
<math display="block">\|x\| := 2 \left|x_1\right| + \sqrt{3 \left|x_2\right|^2 + \max (\left|x_3\right| , 2 \left|x_4\right|)^2}</math>
is a norm on <math>\R^4.</math>

For any norm and any [[Injective function|injective]] [[linear transformation]] <math>A</math> we can define a new norm of <math>x,</math> equal to
<math display="block">\|A x\|.</math>
In 2D, with <math>A</math> a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each <math>A</math> 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 (geometry)|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 <math>\R^n</math> (centered at zero) defines a norm on <math>\R^n</math> (see {{slink||Classification of seminorms: absolutely convex absorbing sets}} below).

All the above formulas also yield norms on <math>\Complex^n</math> without modification.

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

===In abstract algebra===
{{Main|Field norm}}

Let <math>E</math> be a [[finite extension]] of a field <math>k</math> of [[inseparable degree]] <math>p^{\mu},</math> and let <math>k</math> have algebraic closure <math>K.</math>  If the distinct [[Field homomorphism|embeddings]] of <math>E</math> are <math>\left\{\sigma_j\right\}_j,</math> then the '''Galois-theoretic norm''' of an element <math>\alpha \in E</math> is the value <math display=inline>\left(\prod_j {\sigma_k(\alpha)}\right)^{p^{\mu}}.</math>  As that function is homogeneous of degree [[Degree of a field extension|<math>[E : k]</math>]], the Galois-theoretic norm is not a norm in the sense of this article.  However, the <math>[E : k]</math>-th root of the norm (assuming that concept makes sense) is a norm.<ref>{{Cite book|last=Lang|first=Serge|title=Algebra|publisher=Springer Verlag|year=2002|isbn=0-387-95385-X|edition=Revised 3rd|location=New York|pages=284|orig-year=1993}}</ref>

====Composition algebras====

The concept of norm <math>N(z)</math> in [[composition algebra]]s does {{em|not}} share the usual properties of a norm since [[null vector]]s are allowed. A composition algebra <math>(A, {}^*, N)</math> consists of an [[algebra over a field]] <math>A,</math> an [[involution (mathematics)|involution]] <math>{}^*,</math> and a [[quadratic form]] [[Degree of a field extension|<math>N(z) = z z^*</math>]] called the "norm".

The characteristic feature of composition algebras is the [[homomorphism]] property of <math>N</math>: for the product <math>w z</math> of two elements <math>w</math> and <math>z</math> of the composition algebra, its norm satisfies <math>N(wz) = N(w) N(z).</math> In the case of [[division algebra]]s <math>\R,</math> <math>\Complex,</math> <math>\mathbb{H},</math> and <math>\mathbb{O}</math> the composition algebra norm is the square of the norm discussed above. In those cases the norm is a [[definite quadratic form]]. In the [[split algebra]]s the norm is an [[isotropic quadratic form]].

==Properties==

For any norm <math>p : X \to \R</math> on a vector space <math>X,</math> the [[reverse triangle inequality]] holds:
<math display="block">p(x \pm y) \geq |p(x) - p(y)| \text{ for all } x, y \in X.</math>
If <math>u : X \to Y</math> is a continuous linear map between normed spaces, then the norm of <math>u</math> and the norm of the [[transpose]] of <math>u</math> are equal.{{sfn|Trèves|2006|pp=242–243}}

For the [[Lp space|<math>L^p</math>  norms]], we have [[Hölder's inequality]]<ref name="GOLUB">{{cite book|last1=Golub|first1=Gene|title=Matrix Computations|last2=Van Loan|first2=Charles F.|publisher=The Johns Hopkins University Press|year=1996|isbn=0-8018-5413-X|edition=Third|location=Baltimore|page=53|author-link1=Gene H. Golub}}</ref>
<math display="block">|\langle x, y \rangle| \leq \|x\|_p \|y\|_q \qquad \frac{1}{p} + \frac{1}{q} = 1.</math>
A special case of this is the [[Cauchy–Schwarz inequality]]:<ref name="GOLUB" />
<math display="block">\left|\langle x, y \rangle\right| \leq \|x\|_2 \|y\|_2.</math>
[[File:Vector norms.svg|frame|right|Illustrations of [[unit circle]]s in different norms.]]

Every norm is a [[seminorm]] and thus satisfies all [[Seminorm#Algebraic_properties|properties of the latter]]. In turn, every seminorm is a [[sublinear function]] and thus satisfies all [[Sublinear_function#Properties|properties of the latter]]. In particular, every norm is a [[convex function]].

===Equivalence===
<!--[[Equivalent norms]] redirects here-->
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 (geometry)|square]] oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit [[circle]]; while for the infinity norm, it is an axis-aligned square. For any <math>p</math>-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 set|convex]] and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and <math>p \geq 1</math> for a <math>p</math>-norm).

In terms of the vector space, the seminorm defines a [[topology]] on the space, and this is a [[Hausdorff space|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]] of vectors <math>\{v_n\}</math> is said to [[modes of convergence|converge]] in norm to <math>v,</math> if <math>\left\|v_n - v\right\| \to 0</math> as <math>n \to \infty.</math> Equivalently, the topology consists of all sets that can be represented as a union of open [[ball (mathematics)|balls]]. If <math>(X, \|\cdot\|)</math> is a normed space then{{sfn|Narici|Beckenstein|2011|pp=107-113}}
<math>\|x - y\| = \|x - z\| + \|z - y\| \text{ for all } x, y \in X \text{ and } z \in [x, y].</math>

Two norms <math>\|\cdot\|_\alpha</math> and <math>\|\cdot\|_\beta</math> on a vector space <math>X</math> are called '''{{visible anchor|equivalent|Equivalent norms}}''' if they induce the same topology,<ref name="Conrad Equiv norms">{{cite web |url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/equivnorms.pdf |title=Equivalence of norms |last=Conrad |first=Keith |website=kconrad.math.uconn.edu |access-date=September 7, 2020 }}</ref> which happens if and only if there exist positive real numbers <math>C</math> and <math>D</math> such that for all <math>x \in X</math>
<math display="block">C \|x\|_\alpha \leq \|x\|_\beta \leq D \|x\|_\alpha.</math>
For instance, if <math>p > r \geq 1</math> on <math>\Complex^n,</math> then<ref name="Relation between p-norms">{{cite web |url=https://math.stackexchange.com/q/218046 |title=Relation between p-norms|website=Mathematics Stack Exchange}}</ref>
<math display="block">\|x\|_p \leq \|x\|_r \leq n^{(1/r-1/p)} \|x\|_p.</math>

In particular,
<math display="block">\|x\|_2 \leq \|x\|_1 \leq \sqrt{n} \|x\|_2</math>
<math display="block">\|x\|_\infty \leq \|x\|_2 \leq \sqrt{n} \|x\|_\infty</math>
<math display="block">\|x\|_\infty \leq \|x\|_1 \leq n \|x\|_\infty ,</math>
That is,
<math display="block">\|x\|_\infty \leq \|x\|_2 \leq \|x\|_1 \leq \sqrt{n} \|x\|_2 \leq n \|x\|_\infty.</math>
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==
{{Main|Seminorm}}

All seminorms on a vector space <math>X</math> can be classified in terms of [[absolutely convex]] [[Absorbing set|absorbing subset]]s <math>A</math> of <math>X.</math> To each such subset corresponds a seminorm <math>p_A</math> called the '''[[Minkowski functional|gauge]]''' of <math>A,</math> defined as
<math display="block>p_A(x) := \inf \{r \in \R : r > 0, x \in r A\}</math>
where <math>\inf_{}</math> is the [[infimum]], with the property that
<math display="block>\left\{x \in X : p_A(x) < 1\right\} ~\subseteq~ A ~\subseteq~ \left\{x \in X : p_A(x) \leq 1\right\}.</math>
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 <math>(p)</math> of seminorms <math>p</math> that [[separation axiom|separates points]]: the collection of all finite intersections of sets <math>\{p < 1/n\}</math> turns the space into a [[locally convex topological vector space]] so that every p is [[continuous function|continuous]].

Such a method is used to design [[Weak topology|weak and weak* topologies]].

norm case:
:Suppose now that <math>(p)</math> contains a single <math>p:</math> since <math>(p)</math> is [[separation axiom|separating]], <math>p</math> is a norm, and <math>A = \{p < 1\}</math> is its open [[unit ball]]. Then <math>A</math> is an absolutely convex [[Bounded set|bounded]] neighbourhood of 0, and <math>p = p_A</math> is continuous.

:The converse is due to [[Andrey Kolmogorov]]: any locally convex and locally bounded topological vector space is [[normable]]. Precisely:
:If <math>X</math> is an absolutely convex bounded neighbourhood of 0, the gauge <math>g_X</math> (so that <math>X = \{g_X < 1\}</math> is a norm.

==See also==

* {{annotated link|Asymmetric norm}}
* {{annotated link|F-seminorm}}
* {{annotated link|Gowers norm}}
* {{annotated link|Kadec norm}}
* {{annotated link|Least-squares spectral analysis}}
* {{annotated link|Mahalanobis distance}}
* {{annotated link|Magnitude (mathematics)}}
* {{annotated link|Matrix norm}}
* {{annotated link|Minkowski distance}}
* {{annotated link|Minkowski functional}}
* {{annotated link|Operator norm}}
* {{annotated link|Paranorm}}
* {{annotated link|Relation of norms and metrics}}
* {{annotated link|Seminorm}}
* {{annotated link|Sublinear function}}

==References==

{{reflist}}

==Bibliography==

* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!--{{sfn|Bourbaki|1987|p=}}-->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!--{{sfn|Khaleelulla|1982|p=}}-->
* {{Kubrusly The Elements of Operator Theory 2nd Edition 2011}} <!--{{sfn|Kubrusly|2011|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|1999|p=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

{{Banach spaces}}
{{Functional analysis}}
{{Topological vector spaces}}

{{DEFAULTSORT:Norm (Mathematics)}}

[[Category:Functional analysis]]
[[Category:Linear algebra]]
[[Category:Norms (mathematics)| ]]