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.

In Unicode, the representation of the "double vertical line" character is U+2016 DOUBLE VERTICAL LINE. The "double vertical line" symbol should not be confused with the "parallel to" symbol, U+2225 PARALLEL TO, which is intended to denote parallel lines and parallel operators. The double vertical line should also not be confused with U+01C1 ǁ LATIN LETTER LATERAL CLICK, aimed to denote lateral clicks in linguistics.

The single vertical line | has a Unicode representation U+007C | VERTICAL LINE.

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.[7]

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,[8] 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:

This definition is still of some interest for 0 < p < 1, but the resulting function does not define a norm,[9] 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

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 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.

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.