Norm (mathematics)

In 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: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.

This norm can be defined as the square root of the inner product of a vector with itself.

[1] It can also refer to a norm that can take infinite values[2] or to certain functions parametrised by a directed set.

also has the following property: Some authors include non-negativity as part of the definition of "norm", although this is not necessary.

For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation

[9] There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

is a norm on the vector space formed by the real or complex numbers.

is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces

This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem.

However, all these norms are equivalent in the sense that they all define the same topology on finite-dimensional spaces.

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

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

but the resulting function does not define a norm,[13] because it violates the triangle inequality.

However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms.

The F-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.

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.

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.

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).

in composition algebras does not share the usual properties of a norm since null vectors are allowed.

The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets.

Equivalently, the topology consists of all sets that can be represented as a union of open balls.

are called equivalent if they induce the same topology,[8] which happens if and only if there exist positive real numbers

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.