Encyclopedia
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
seminorm 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 are usually drawn as arrows in a 2-dimensional
cartesian coordinate system starting at the origin . The Euclidean norm assigns to each vector the length of its arrow.
A vector space with a norm is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space.
Definition
Given a vector space
V over a subfield
F of the
complex numbers such as the complex numbers themselves or the real or rational numbers, a
seminorm on V is a function
p:
V?
R;
x?
p with the following properties:
For all
a in
F and all
u and
v in
V,
- p = 0
- p = |a| p,
- p = p + p .
Positivity is a simple consequence of the two axioms, positive homogeneity and the triangle inequality.
A
norm is a
seminorm with the additional property
- p = 0 if and only if v is the zero vector .
A norm is usually denoted ||
v||, and sometimes |
v|, instead of
p.
Although every vector space is seminormed , it may not be normed. Any vector space
V with seminorm
p can be made into a normed space by forming the
quotient space V/W where
W is the subspace of
V consisting of all vectors
v in
V with
p = 0. The induced norm on
V/W is given by ||
W+
v|| =
p and is clearly well-defined.
A topological vector space is called
normable if the
topology of the space can be induced by a norm .
Properties
for all
u and
v ?
V:
- p = | p - p |
Examples
- All norms are seminorms.
- The trivial seminorm, with p = 0 for all x in V.
- The absolute value is a norm on the real numbers.
- Every linear form f on a vector space defines a seminorm by x?|f|.
Euclidean norm
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.
On
Cn the most common norm is
, equivalent with the Euclidean norm on
R2n.
In each case we can also express the norm as the square root of the
inner product of the vector and itself. The euclidean norm is also called the
l 2, see
Lp space.
The set of vectors whose Euclidean norm is a given constant forms the surface of a
sphere.
Taxicab norm or Manhattan norm
Main article Taxicab geometryThe name relates to the distance a taxi has to drive in a rectangular
street grid to get from the origin to the point
x.
The set of vectors whose 1-norm is a given constant forms the surface of a
cross polytope.
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
Lp space.
Infinity norm or maximum norm
Main article maximum normThe set of vectors whose 8-norm is a given constant forms the surface of a
measure polytope.
Zero norm
In the machine learning and optimization literature, one often finds reference to the zero norm. The zero norm of
x is defined as where is the
p-norm defined above. If we define then we can write the zero norm as . It follows that the zero norm of
x is simply the number of non-zero elements of
x. Despite its name, the zero norm is
not a true norm; in particular, it is not positive homogeneous.
Other norms
Other norms on
Rn can be constructed by combining the above; for example
is a norm on
R4.
For any norm and any
bijective linear transformation
A we can define a new norm of
x, equal to
In 2D, with
A a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. In 2D, each
A applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a
parallelogram of a particular shape, size and orientation. In 3D this is similar but different for the 1-norm and the maximum norm .
All the above formulas also yield norms on
Cn without modification.
Infinite dimensional case
The generalization of the above norms to an infinite number of components leads to the
Lp spaces, with norms
resp.
, which can be further generalized .
Any
inner product induces in a natural way the norm
Other examples of infinite dimensional normed vector spaces can be found in the Banach space article.
Properties
The concept of
unit circle is different in different norms: for the 1-norm the unit circle in
R2 is a
rhomboid, for the 2-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. On a finite dimensional vector space all norms are equivalent.
Equivalent 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 -norm is a sublinear function, which implies that every norm is a
convex function. As a result, finding a global optimum of a norm-based objective function is often tractable.
Given a finite family of seminorms
pi on a vector space the sum
is again a seminorm.
Absolutely convex and absorbing sets
Seminorms are closely related to
absolutely convex and absorbing sets. Let
p be a seminorm on a vector space
V, then for any scalar a the sets and are absorbing and absolutely convex. In a normed vector space the set is called the
closed unit ball.
Conversely to each absorbing and absolutely convex subset
A of
V corresponds a seminorm
p called the
gauge of
A, defined as
- p := inf
with the property that
- ? A ? .
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 seminorms. Typically this family is infinite, and there are enough seminorms to distinguish between elements of the vector space, creating a Hausdorff space.
See also
- inner product, a vector multiplication which induces a norm
- relation of norms and metrics - a translation invariant and homogeneous metric can be used to define a norm
- normed vector space
- matrix norm
- Manhattan distance
