In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly 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 nonzero vectors.
A simple example is the 2dimensional Euclidean space R^{2} equipped with the Euclidean norm. Elements in this vector space (e.g., (3, 7) ) are usually drawn as arrows in a 2dimensional cartesian coordinate system starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude.
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.
Contents 
Given a vector space V over a subfield F of the complex numbers, including imaginary numbers or real numbers, a norm on V is a function with the following properties:
For all a in F and all u and v in V,
A simple consequence of the first two axioms, positive homogeneity and the triangle inequality, is p(0) = 0 and thus
A seminorm is a norm with the requirement of positive definiteness removed.
Although every vector space is seminormed (e.g., with the trivial seminorm in the Examples section below), it may be not normed. Every vector space V with seminorm p(v) induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is clearly welldefined and is given by:
A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm).
The norm of a vector v is usually denoted v, and sometimes v. However, the latter notation is generally discouraged, because it is also used to denote the absolute value of scalars and the determinant of matrices.
On R^{n}, the intuitive notion of length of the vector x = [x_{1}, x_{2}, ..., 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 R^{n}, but there are other norms on this vector space as will be shown below. However all these norms are equivalent in the sense that they all define the same topology.
On C^{n} the most common norm is
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 Euclidean length and the L_{2} distance or L^{2} norm; see L^{p} space.
The set of vectors whose Euclidean norm is a given constant forms the surface of an nsphere, with n+1 being the dimension of the Euclidean space.
The 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 1norm is a given constant forms the surface of a cross polytope of dimension equivalent to that of the norm minus 1. The Taxicab norm is also called the L^{1} norm. The distance derived from this norm is called the Manhattan distance or L_{1} distance.
In contrast,
is not a norm because it may yield negative results.
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.
This formula is also valid for 0 < p < 1, but the resulting function does not define a norm,^{[1]} because it violates the triangle inequality.
Taking the limit yields the maximum norm.
The set of vectors whose infinity norm is a given constant, c, forms the surface of a hypercube with edge length 2c.
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 pnorm defined above. If we define 0^{0}: = 0 then we can write the zero norm as . It follows that the zero norm of x is simply the number of nonzero elements of x. Despite its name, the zero norm is not a true norm; in particular, it is not positive homogeneous. Such a norm can be defined over arbitrary fields (besides the fields of complex numbers). In the context of the information theory, it is often called the Hamming distance in the case of the 2element GF(2) field.
Other norms on R^{n} can be constructed by combining the above; for example
is a norm on R^{4}.
For any norm and any injective 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 1norm (octahedrons) and the maximum norm (prisms with parallelogram base).
All the above formulas also yield norms on C^{n} without modification.
The generalization of the above norms to an infinite number of components leads to the L^{p} spaces, with norms
(for complexvalued sequences x resp. functions f defined on ), which can be further generalized (see Haar measure).
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.
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1norm the unit circle in R^{2} is a square, for the 2norm (Euclidean norm) it is the wellknown unit circle, while for the infinity norm it is a different square. For any pnorm it is a superellipse (with congruent axes). See the accompanying illustration. Note that due to the definition of the norm, the unit circle is always convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle).
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. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors {v_{n}} is said to converge in norm to v if as . Equivalently, the topology consists of all sets that can be represented as a union of open balls.
Two norms •_{α} and •_{β} on a vector space V are called equivalent if there exist positive real numbers C and D such that
for all x in V. For instance, on :
If the vector space is a finitedimensional real/complex one, all norms are equivalent. If not, some norms are not.
Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. 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, which implies that every norm is a convex function. As a result, finding a global optimum of a normbased objective function is often tractable.
Given a finite family of seminorms p_{i} on a vector space the sum
is again a seminorm.
For any norm p on a vector space V, we have that for all u and v ∈ V:
For the l^{p} norms, we have the Hölder's inequality^{[2]}
A special case of the above property is the CauchySchwarz inequality:^{[2]}
All seminorms on a vector space V can be classified in terms of absolutely convex absorbing sets in V. To each such set, A, corresponds a seminorm p_{A} called the gauge of A, defined as
with the property that
Conversely:
Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common method to construct such a basis is to use a separating family (p) of seminorms p: the collection of all finite intersections of sets {p<1/n} turns the space into a locally convex topological vector space so that every p is continuous.
A such method is used to design weak and weak* topologies.
norm case:
