Norm（范数）

(from:wikipedia)

In linear algebra, functional analysis and related areas ofmathematics, a norm is a function that assigns a strictly positivelength or size to all vectors in a vector space, other than the zero vector (which has zero length assigned to it). A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).

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 systemstarting 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.

Notation

The norm of a vector, matrix, or set (its cardinality) is usually noted using the "double vertical line", Unicode Ux2016 : ( ‖ ). For example, the norm of a vector v is usually denoted ‖v‖. Sometimes the vertical line, Unicode Ux007c ( | ), is used (e.g. |v|), but this latter notation is generally discouraged, because it is also used to denote the absolute value of scalars and the determinant of matrices. The double vertical line should not be confused with the "parallel to" symbol, Unicode Ux2225 ( ∥ ). This is usually not a problem because ‖ is used in parenthesis-like fashion, whereas ∥ is used as an infix operator.

Definition

Given a vector space V over a subfield F of the complex numbers, a norm on V is a function p: V → R with the following properties:

For all a ∈ F and all u, v ∈ V,

1. p(av) = |a| p(v), (positive homogeneity or positive scalability).
2. p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).
3. If p(v) = 0 then v is the zero vector (separates points).

A simple consequence of the first two axioms, positive homogeneity and the triangle inequality, is p(0) = 0 and thus

p(v) ≥ 0 (positivity).

A seminorm is a norm with the 3rd property (separating points) removed.

Although every vector space is seminormed (e.g., with the trivial seminorm in the Examples section below), it may not be 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 well-defined and is given by:

p(W + v) = p(v).

A topological vector space is called normable (seminormable) if thetopology of the space can be induced by a norm (seminorm).