Norm (mathematics)

In mathematics, 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.[1] 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.

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

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 vV,

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]

In Unicode, 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 in linguistics.

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 x = (xi)iI is a Hamel basis for a vector space X then the real-valued map that sends x = iI sixiX (where all but finitely many of the scalars si are 0) to iI |si| is a norm on X.[5] There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

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

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 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

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

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

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:

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 1 norm. The distance derived from this norm is called the Manhattan distance or 1 distance.

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:

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 Lp class is a vector space, and it is also true that the function

(without pth root) defines a distance that makes Lp(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 set of vectors whose infinity norm is a given constant, c, forms the surface of a hypercube with edge length 2c.

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[who?] 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 p and Lp spaces, with norms

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

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

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

For any norm p on a vector space V, the reverse triangle inequality holds: for all u and vV,

If u : XY is a continuous linear map between normed space, then the norm of u and the norm of the transpose of u are equal.[13]

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

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.

All seminorms on a vector space V can be classified in terms of absolutely convex absorbing subsets A of V. To each such subset corresponds a seminorm pA called the gauge of A, defined as