(from:wikipedia)
In , and related areas of, a norm is a that assigns a strictly positivelength or size to all in a , other than the (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 R2 equipped with the . Elements in this vector space (e.g., (3, 7)) are usually drawn as arrows in a 2-dimensional 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 .
A vector space with a norm is called a . Similarly, a vector space with a seminorm is called a seminormed vector space.
Notation
The norm of a vector, , or (its ) 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 of scalars and the 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 .
Definition
Given a V over a F of the , a norm on V is a p: V → R with the following properties:
For all a ∈ F and all u, v ∈ V,
- p(av) = |a| p(v), ( or positive scalability).
- p(u + v) ≤ p(u) + p(v) ( or ).
- If p(v) = 0 then v is the (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 , 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 is called normable (seminormable) if the of the space can be induced by a norm (seminorm).