Norm (mathematics) Guide, Meaning , Facts, Information and Description

Table of contents

1 Definition 2 Examples of norms 2.1 Euclidean norm 2.2 Taxicab norm or Manhattan norm 2.3 p-norm 2.4 Infinity norm or maximum norm 2.5 Other norms

Definition

Given a vector space V over K, a norm on V is a function ||·||:V->R; x->||x|| with the following properties:

For all a ∈ K and all u and v ∈ V,

1. ||v|| ≥ 0, with equality if and only if v = 0,

2. ||av|| = |a| ||v||,

3. ||u + v|| ≤ ||u|| + ||v||.

Most of property 1 follows from the other axioms, and in fact it can be replaced by the following condition:

1'. If ||v|| = 0, then v = 0.

A useful consequence of the norm axioms is the inequality

||u ± v|| ≥ | ||u|| - ||v|| |

for all u and v ∈ K.

Examples of norms

Euclidean norm

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

This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. The Euclidean norm is by far the most commonly used norm on Rⁿ, but there are other norms on this vector space as will be shown below.

Taxicab norm or Manhattan norm

Main article Taxicab geometry

The name comes from the fact that the norm gives the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x.

Illustrations of unit circles in different norms.

p-norm

Let p≥1 be a real number.

Note that for p=1 we get the taxicab norm and for p=2 we get the Euclidean norm. See also L^p space.

Infinity norm or maximum norm

Main article maximum norm

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 in R² is a rhomboid, for the 2-norm (Euclidean norm) it is the well-known unit circle, while for the infinity norm it is a square. See the accompanying illustration.

Other norms

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

is a norm on R⁴.

All the above formulas also yield norms on Cⁿ without modification.

Examples of infinite dimensional normed vector spaces can be found in the Banach space article. In addition, inner product space becomes a normed vector space if we define the norm as

This is an Article on Norm (mathematics). Page Contains Information, Facts Details or Explanation Guide About Norm (mathematics)

	Web www.E-paranoids.com