Every (real or complex) vector space admits a norm: If x_{\bull} = \left(x_i\right)_{i \in I} is a
Hamel basis for a vector space X then the real-valued map that sends x = \sum_{i \in I} s_i x_i \in X (where all but finitely many of the scalars s_i are 0) to \sum_{i \in I} \left|s_i\right| is a norm on X. 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 |x| 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 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 : \mathbb{F} \to X, where \mathbb{F} is either \R or \Complex, and norm-preserving means that |x| = p(f(x)). This isomorphism is given by sending 1 \isin \mathbb{F} to a vector of norm 1, which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.
Euclidean norm On the n-dimensional
Euclidean space \R^n, the intuitive notion of length of the vector \boldsymbol{x} = \left(x_1, x_2, \ldots, x_n\right) is captured by the formula \|\boldsymbol{x}\|_2 := \sqrt{x_1^2 + \cdots + x_n^2}. 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 \R^n,
\ell^2 norm,
2-norm, or
square norm; see
L^p space. It defines a
distance function called the
Euclidean length,
L^2 distance, or
\ell^2 distance. The set of vectors in \R^{n+1} whose Euclidean norm is a given positive constant forms an
n-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 \R^2. This identification of the complex number x + i y as a vector in the Euclidean plane, makes the quantity \sqrt{x^2 + y^2} (as first suggested by Euler) the Euclidean norm associated with the complex number. For z = x +iy, the norm can also be written as \sqrt{\bar z z} where \bar z is the
complex conjugate of z\,.
Quaternions and octonions There are exactly four
Euclidean Hurwitz algebras over the
real numbers. These are the real numbers \R, the complex numbers \Complex, the
quaternions \mathbb{H}, and lastly the
octonions \mathbb{O}, where the dimensions of these spaces over the real numbers are 1, 2, 4, \text{ and } 8, respectively. The canonical norms on \R and \Complex are their
absolute value functions, as discussed previously. The canonical norm on \mathbb{H} of
quaternions is defined by \lVert q \rVert = \sqrt{\,qq^*~} = \sqrt{\,q^*q~} = \sqrt{\, a^2 + b^2 + c^2 + d^2 ~} for every quaternion q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k in \mathbb{H}. This is the same as the Euclidean norm on \mathbb{H} considered as the vector space \R^4. Similarly, the canonical norm on the
octonions is just the Euclidean norm on \R^8.
Finite-dimensional complex normed spaces On an n-dimensional
complex space \Complex^n, the most common norm is \|\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}. In this case, the norm can be expressed as the
square root of the
inner product of the vector and itself: \|\boldsymbol{x}\| := \sqrt{\boldsymbol{x}^H ~ \boldsymbol{x}}, where \boldsymbol{x} is represented as a
column vector \begin{bmatrix} x_1 \; x_2 \; \dots \; x_n \end{bmatrix}^{\rm T} and \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: \|\boldsymbol{x}\| := \sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}.
Taxicab norm or Manhattan norm \|\boldsymbol{x}\|_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 (like that of the
New York borough of
Manhattan) 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, which has dimension equal to the dimension of the vector space minus 1. The Taxicab norm is also called the
\ell^1 norm. The distance derived from this norm is called the
Manhattan distance or
\ell^1 distance. The 1-norm is simply the sum of the absolute values of the columns. In contrast, \sum_{i=1}^n x_i is not a norm because it may yield negative results.
p-norm Let p \geq 1 be a real number. The p-norm (also called \ell^p-norm) of vector \mathbf{x} = (x_1, \ldots, x_n) is because it violates the
triangle inequality. What is true for this case of 0 even in the measurable analog, is that the corresponding L^p class is a vector space, and it is also true that the function \int_X |f(x) - g(x)|^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 \frac{\partial}{\partial x_k} \|\mathbf{x}\|_p = \frac{x_k \left|x_k\right|^{p-2}} { \|\mathbf{x}\|_p^{p-1}}. The derivative with respect to x, therefore, is \frac{\partial \|\mathbf{x}\|_p}{\partial \mathbf{x}} =\left(\frac{\mathbf{x} \circ |\mathbf{x}|^{p-2}} {\|\mathbf{x}\|^{p-1}_p}\right)^\top. where \circ denotes
Hadamard product and |\cdot| is used for absolute value of each component of the vector. For the special case of p = 2, this becomes \frac{\partial}{\partial x_k} \|\mathbf{x}\|_2 = \frac{x_k}{\|\mathbf{x}\|_2}, or \frac{\partial}{\partial \mathbf{x}} \|\mathbf{x}\|_2 = \left(\frac{\mathbf{x}}{ \|\mathbf{x}\|_2}\right)^\top.
Maximum norm (special case of: infinity norm, uniform norm, or supremum norm) If \mathbf{x} is some vector such that \mathbf{x} = (x_1, x_2, \ldots ,x_n), then: \|\mathbf{x}\|_\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 2 c.
Energy norm The energy norm of a vector \boldsymbol{x} = \left(x_1, x_2, \ldots, x_n\right) \in \R^{n} is defined in terms of a
symmetric positive definite matrix A \in \R^n as {\|\boldsymbol{x}\|}_{A} := \sqrt{\boldsymbol{x}^{T} \cdot A \cdot \boldsymbol{x}}. It is clear that if A is the
identity matrix, this norm corresponds to the
Euclidean norm. If A is diagonal, this norm is also called a
weighted norm. The energy norm is induced by the
inner product given by \langle \boldsymbol{x}, \boldsymbol{y} \rangle_A := \boldsymbol{x}^{T} \cdot A \cdot \boldsymbol{y} for \boldsymbol{x}, \boldsymbol{y} \in \R^{n}. In general, the value of the norm is dependent on the
spectrum of A: For a vector \boldsymbol{x} with a Euclidean norm of one, the value of {\|\boldsymbol{x}\|}_{A} is bounded from below and above by the smallest and largest absolute
eigenvalues of A respectively, where the bounds are achieved if \boldsymbol{x} coincides with the corresponding (normalized) eigenvectors. Based on the symmetric
matrix square root A^{1/2}, the energy norm of a vector can be written in terms of the standard Euclidean norm as {\|\boldsymbol{x}\|}_{A} = {\|A^{1/2} \boldsymbol{x}\|}_{2}.
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 (x_n) \mapsto \sum_n{2^{-n} x_n/(1+x_n)}. Here we mean by
F-norm some real-valued function \lVert \cdot \rVert on an F-space with distance d, such that \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.
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 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-non-zeros function the L^0 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
\ell^p and L^p spaces for p \ge 1\,, with norms \|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} for complex-valued sequences and functions on X \sube \R^n respectively, which can be further generalized (see
Haar measure). These norms are also valid in the limit as p \rightarrow +\infty, giving a
supremum norm, and are called \ell^\infty and L^\infty\,. Any
inner product induces in a natural way the norm \|x\| := \sqrt{\langle x , x\rangle}. 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 \ell^p space gives a
strictly finer topology than an infinite-dimensional \ell^q space when p
Composite norms Other norms on \R^n can be constructed by combining the above; for example \|x\| := 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 \R^4. For any norm and any
injective linear transformation A we can define a new norm of x, equal to \|A x\|. 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 \R^n (centered at zero) defines a norm on \R^n (see below). All the above formulas also yield norms on \Complex^n without modification. There are also norms on spaces of matrices (with real or complex entries), the so-called
matrix norms.
In abstract algebra Let E be a
finite extension of a field k of
inseparable degree p^{\mu}, and let k have algebraic closure K. If the distinct
embeddings of E are \left\{\sigma_j\right\}_j, then the
Galois-theoretic norm of an element \alpha \in E is the value \left(\prod_j {\sigma_k(\alpha)}\right)^{p^{\mu}}. As that function is homogeneous 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.
Composition algebras The concept of norm N(z) in
composition algebras does share the usual properties of a norm since
null vectors are allowed. A composition algebra (A, {}^*, N) consists of an
algebra over a field A, an
involution {}^*, and a
quadratic form N(z) = z z^* called the "norm". The characteristic feature of composition algebras is the
homomorphism property of N: for the product w z of two elements w and z of the composition algebra, its norm satisfies N(wz) = N(w) N(z). In the case of
division algebras \R, \Complex, \mathbb{H}, and \mathbb{O} 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 algebras the norm is an
isotropic quadratic form. ==Properties==