Norm (mathematics)

Suppose that p and q are two norms (or seminorms) on a vector space V. Then p and q are called equivalent, if there exist two real constants c and C with c > 0 such that for every vector v ∈ V,

${\displaystyle cq(\mathbf {v} )\leq p(\mathbf {v} )\leq Cq(\mathbf {v} ).}$

The norms p and q are equivalent if and only if they induce the same topology on V.[4] Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.[4]

The absolute value

${\displaystyle \left\|x\right\|=\left|x\right|}$

is a norm on the one-dimensional vector spaces formed by the real or complex numbers.[6]

Any norm p on a one-dimensional vector space X is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces f : 𝔽 → X, where 𝔽 is either ℝ or ℂ, and norm-preserving means that ${\displaystyle \left|x\right|=p(f(x))}$. This isomorphism is given by sending 1 ∈ 𝔽 to a vector of norm 1, which exists since such a vector is obtained by multiplying any nonzero vector by the inverse of its norm.

On the n-dimensional Euclidean space ℝⁿ, the intuitive notion of length of the vector x = (x₁, x₂, ..., x_n) is captured by the formula

${\displaystyle \left\|{\boldsymbol {x}}\right\|_{2}:={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}.}$[7]

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.[8]

The Euclidean norm is by far the most commonly used norm on ℝⁿ,[7] 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

${\displaystyle \left\|{\boldsymbol {x}}\right\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}$

The Euclidean norm is also called the L² norm,[9] ℓ² norm, 2-norm, or square norm; see L^p space. It defines a distance function called the Euclidean length, L² distance, or ℓ² distance.

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

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 ℝ². This identification of the complex number x + i y as a vector in the Euclidean plane, makes the quantity ${\displaystyle {\sqrt {x^{2}+y^{2}}}}$ (as first suggested by Euler) the Euclidean norm associated with the complex number.

There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers ℝ, the complex numbers ℂ, the quaternions ℍ, and lastly the octonions 𝕆, where the dimensions of these spaces over the real numbers are 1, 2, 4, and 8, respectively. The canonical norms on ℝ and ℂ are their absolute value functions, as discussed previously.

The canonical norm on ℍ of quaternions is defined by

${\displaystyle \lVert q\rVert ={\sqrt {\,qq^{*}~}}={\sqrt {\,q^{*}q~}}={\sqrt {\,a^{2}+b^{2}+c^{2}+d^{2}~}}}$

for every quaternion ${\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }$ in ℍ. This is the same as the Euclidean norm on ℍ considered as the vector space ℝ⁴. Similarly, the canonical norm on the octonions is the just Euclidean norm on ℝ⁸.

Finite-dimensional complex normed spaces

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

${\displaystyle \left\|{\boldsymbol {z}}\right\|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}_{1}+\cdots +z_{n}{\bar {z}}_{n}}}.}$

In this case, the norm can be expressed as the square root of the inner product of the vector and itself:

${\displaystyle \left\|{\boldsymbol {x}}\right\|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},}$

where ${\displaystyle {\boldsymbol {x}}}$ is represented as a column vector ([x₁; x₂; ...; x_n]), and ${\displaystyle {\boldsymbol {x}}^{H}}$ 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:

${\displaystyle \left\|{\boldsymbol {x}}\right\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}$

${\displaystyle \left\|{\boldsymbol {x}}\right\|_{1}:=\sum _{i=1}^{n}\left|x_{i}\right|.}$

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 ${\displaystyle \ell }$₁ norm. The distance derived from this norm is called the Manhattan distance or ${\displaystyle \ell }$₁ distance.

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

In contrast,

${\displaystyle \sum _{i=1}^{n}x_{i}}$

is not a norm because it may yield negative results.

Let p ≥ 1 be a real number. The p-norm (also called ${\displaystyle \ell _{p}}$-norm) of vector ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ is

${\displaystyle \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}.}$[7]

For p = 1, we get the taxicab norm,[6] for p = 2, we get the Euclidean norm, and as p approaches ∞ the p-norm approaches the infinity norm or maximum norm:

${\displaystyle \left\|\mathbf {x} \right\|_{\infty }:=\max _{i}\left|x_{i}\right|.}$

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,[10] 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 L^p class is a vector space, and it is also true that the function

${\displaystyle \int _{X}\left|f(x)-g(x)\right|^{p}~\mathrm {d} \mu }$

(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, 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 p-norm is given by

${\displaystyle {\frac {\partial }{\partial x_{k}}}\left\|\mathbf {x} \right\|_{p}={\frac {x_{k}\left|x_{k}\right|^{p-2}}{\left\|\mathbf {x} \right\|_{p}^{p-1}}}.}$

The derivative with respect to x, therefore, is

${\displaystyle {\frac {\partial \|\mathbf {x} \|_{p}}{\partial \mathbf {x} }}={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.}$

where ∘ denotes Hadamard product and ${\displaystyle |\cdot |}$ is used for absolute value of each component of the vector.

For the special case of p = 2, this becomes

${\displaystyle {\frac {\partial }{\partial x_{k}}}\left\|\mathbf {x} \right\|_{2}={\frac {x_{k}}{\left\|\mathbf {x} \right\|_{2}}},}$

or

${\displaystyle {\frac {\partial }{\partial \mathbf {x} }}\left\|\mathbf {x} \right\|_{2}={\frac {\mathbf {x} }{\left\|\mathbf {x} \right\|_{2}}}.}$

${\displaystyle \left\|x\right\|_{\infty }=1}$

If ${\displaystyle \mathbf {x} }$ is some vector such that ${\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})}$, then:

${\displaystyle \left\|\mathbf {x} \right\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}$

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

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 ${\displaystyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}}$.[11] Here we mean by F-norm some real-valued function ${\displaystyle \lVert \ \cdot \ \rVert }$ on an F-space with distance d, such that ${\displaystyle \lVert x\rVert =d(x,0)}$. The F-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.

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.

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

${\displaystyle \left\|x\right\|_{p}={\bigg (}\sum _{i\in \mathbb {N} }\left|x_{i}\right|^{p}{\bigg )}^{1/p}{\text{ and }}\ \left\|f\right\|_{p,X}={\bigg (}\int _{X}\left|f(x)\right|^{p}~\mathrm {d} x{\bigg )}^{1/p}}$

for complex-valued sequences and functions on X ⊆ ℝⁿ respectively, which can be further generalized (see Haar measure).

Any inner product induces in a natural way the norm ${\displaystyle \left\|x\right\|:={\sqrt {\langle x,x\rangle }}.}$

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

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

${\displaystyle \left\|x\right\|:=2\left|x_{1}\right|+{\sqrt {3\left|x_{2}\right|^{2}+\max(\left|x_{3}\right|,2\left|x_{4}\right|)^{2}}}}$

is a norm on ℝ⁴.

For any norm and any injective linear transformation A we can define a new norm of x, equal to

${\displaystyle \left\|Ax\right\|.}$

In 2D, with A a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. 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 (see § Classification of seminorms: absolutely convex absorbing sets below).

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.

Let E be a finite extension of a field k of inseparable degree p^μ, and let k have algebraic closure K. If the distinct embeddings of E are {σ_j}_j, then the Galois-theoretic norm of an element α∈E is the value ${\displaystyle \left(\prod _{j}{\sigma _{k}(\alpha )}\right)^{p^{\mu }}}$. As that function is homogenous of degree [E:k], the Galois-theoretic norm is not a norm in the sense of this article. However, the [E:k]^th root of the norm (assuming that concept makes sense), is a norm.[12]

The concept of norm ${\displaystyle N(z)}$ in composition algebras 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 ${\displaystyle N(z)=zz^{*},}$ 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 ${\displaystyle N(wz)=N(w)N(z).}$ For ℝ, ℂ, ℍ, 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.

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 ${\displaystyle p\geq 1}$ 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 of vectors ${\displaystyle \{v_{n}\}}$ is said to converge in norm to ${\displaystyle v}$, if ${\displaystyle \left\|v_{n}-v\right\|\rightarrow 0}$ as ${\displaystyle n\to \infty }$. Equivalently, the topology consists of all sets that can be represented as a union of open balls. If (X, ||⋅||) is a normed space then ||x - y|| = ||x - z|| + ||z - y|| for all x, y, z ∈ X.[15]

Two norms ‖•‖_α and ‖•‖_β on a vector space V are called equivalent if they induce the same topology,[4] which happens if and only if there exist positive real numbers C and D such that for all x in V

${\displaystyle C\left\|x\right\|_{\alpha }\leq \left\|x\right\|_{\beta }\leq D\left\|x\right\|_{\alpha }.}$

For instance, if p > r ≥ 1 on ${\displaystyle \mathbf {C} ^{n}}$, then

${\displaystyle \left\|x\right\|_{p}\leq \left\|x\right\|_{r}\leq n^{(1/r-1/p)}\left\|x\right\|_{p}}$[16]

In particular,

${\displaystyle \left\|x\right\|_{2}\leq \left\|x\right\|_{1}\leq {\sqrt {n}}\left\|x\right\|_{2}}$

${\displaystyle \left\|x\right\|_{\infty }\leq \left\|x\right\|_{2}\leq {\sqrt {n}}\left\|x\right\|_{\infty }}$

${\displaystyle \left\|x\right\|_{\infty }\leq \left\|x\right\|_{1}\leq n\left\|x\right\|_{\infty },}$

That is,

${\displaystyle \left\|x\right\|_{\infty }\leq \left\|x\right\|_{2}\leq \left\|x\right\|_{1}\leq {\sqrt {n}}\left\|x\right\|_{2}\leq n\left\|x\right\|_{\infty }.}$

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.