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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.
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)
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.
- 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.
On Rn, the intuitive notion of length of the vector x = (x1, x2, ..., xn) 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.
p-normLet 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 Lp space.
Infinity norm or maximum norm
Main article maximum norm
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.
|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 pi on a vector space the sum
is again a semi-norm.
A characterisation of locally convex topological vector spaces has the topology defined by a family of seminorms, and many examples use this characterisation in their definitions. Typically this family is infinite, and there are enough seminorms to distinguish between elements of the vector space, creating a Hausdorff space.
- inner product, a vector multiplication which induces a norm
- metric (mathematics), a translation invariant metric can be used to define a norm
- normed vector space
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