Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. A semi norm on the other hand is allowed to assign zero length to some non-zero vectors.

A simple example is the 2-dimensional Euclidean space R² equipped with the Euclidean norm. Elements in this vector space (e.g. (3,7) ) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin (0,0). The Euclidean norm assigns to each vector the length of its arrow.

A vector space with a norm (semi-norm) is called a normed vector space (semi normed vector space).

1 Definition
2 Notes
3 Examples
4 Properties
5 Absolutely convex and absorbing sets
6 See also

Definition

Given a vector space V over a field F (the real or complex numbers) a semi-norm on V is a function p:V->R; x-> p(x) with the following properties:

For all a ∈ F and all u and v ∈ V,

p(v) ≥ 0 (positivity)
p(a v) = |a| p(v), (positive homogenity or positive scalability)
p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).

A norm is a semi-norm with the additional property

p(v) = 0 if and only if v is the zero vector (positive definite)

A topological vector space is called normable (semi-normable) if the topology of the space can be induced by a norm (semi norm).

Notes

Semi norms are often denoted by p(v) whereas norms are traditionally denoted ||v||. If you are new to mathematics don't be confused by the norm being denoted by ||.|| rather than a letter (as usual for functions) and by the image of an element x of the domain under the norm being denoted by ||x|| rather than ||.||(x) which would be the usual notation for a function denoted ||.||.

A useful consequence of the norm axioms is the inequality

||u ± v|| ≥ | ||u|| − ||v|| |

for all u and v ∈ K.

Examples

All norms are semi norms
The trivial seminorms — those where p(x) = 0 for all x in V.
The absolute value is a norm on the real numbers.

Euclidean norm

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

<math>\|x\| := \sqrt{|x_1|^2 + \cdots + |x_n|^2}.<math>

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.

p-norm

Let p≥1 be a real number.

<math>\|x\|_p := \left( \sum_{i=1}^n |x_i|^p \right)^\frac{1}{p}<math>

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

<math>\|x\|_\infty := \max \left(|x_1|, \ldots ,|x_n| \right).<math>

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

Properties

Illustrations of unit circles in different norms.

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.

In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm.

Two norms ||·||₁ and ||·||₂ on a vector space V are called equivalent if there exist positive real numbers C and D such that

for all x in V. Equivalend norms define the same notions of continuity and convergence and do not need to be distinguished for most purposes. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic.

Every (semi)-norm is a sublinear function.

Given a finite familiy of semi-norms p_i on a vector space the sum

is again a semi-norm.

Absolutely convex and absorbing sets

Semi norms are closely related to absolutely convex and absorbing sets. Let p be a semi norm on a vector space V, then for any scalar α the sets {x : p(x) < α} and {x : p(x) ≤ α} are absorbing and absolutely convex. In a normed vector space the set {x : p(x) ≤ 1} is called unit ball.

Conversely to each absorbing and absolutely convex subset A of V corresponds a semi norm p called the gauge of A, defined as

p(x) := inf{α : α > 0, x ∈ α A}

with the property that

{x : p(x) < 1} ⊆ A ⊆ {x : p(x) ≤ 1}.

A locally convex topological vector space has a local basis consisting of absolutely convex and absorbing sets. A common method to construct such a basis is to use a familiy of semi norms. Typically this family is infinite, and there are enough seminorms to distinguish between elements of the vector space, creating a Hausdorff space.