|
In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive ''length'' or ''size'' to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. A simple example is the 2-dimensional Euclidean space R2 equipped with the Euclidean norm. Elements in this vector space (e.g., ) 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. Because of this, the Euclidean norm is often known as the magnitude. A vector space on which a norm is defined is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a given vector space in more than one way. ==Definition== Given a vector space ''V'' over a subfield ''F'' of the complex numbers, a norm on ''V'' is a function with the following properties: For all ''a'' ∈ ''F'' and all u, v ∈ ''V'', # ''p''(''a''v) = ''p''(v), (''absolute homogeneity'' or ''absolute scalability''). # ''p''(u + v) ≤ ''p''(u) + ''p''(v) (''triangle inequality'' or ''subadditivity''). # If ''p''(v) = 0 then v is the zero vector (''separates points''). By the first axiom, absolute homogeneity, we have and , so that by the triangle inequality : ''p''(v) ≥ 0 (''non-negativity''). A seminorm on ''V'' is a function with the properties 1. and 2. above. Every vector space ''V'' with seminorm ''p'' 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 . The induced norm on ''V''/''W'' is clearly well-defined and is given by: : ''p''(''W'' + v) = ''p''(v). Two norms (or seminorms) ''p'' and ''q'' on a vector space ''V'' are equivalent if there exist two real constants ''c'' and ''C'', with such that :for every vector v in ''V'', one has that: . A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Norm (mathematics)」の詳細全文を読む スポンサード リンク
|