Norm (mathematics)
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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, ..., xn) 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
- <math>\|x\|_1 = \sum_{i=1}^{n} |x_i|.<math>
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.- <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 Lp space.
Infinity norm or maximum norm
Main article maximum norm
- <math>\|x\|_\infty = \max \left(|x_1|, \ldots ,|x_n| \right).<math>
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
- <math>\|x\| = 2|x_1| + \sqrt{3|x_2|^2 + \max(|x_3|,2|x_4|)^2}<math>
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
- <math>\|x\| = \sqrt{
