a norm is a function that assigns a strictly positive length or size to each vector 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 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., (3, 7)) are usually drawn as arrows in a 2-dimensional 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 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.
http://mathworld.wolfram.com/Norm.html
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., (3, 7)) are usually drawn as arrows in a 2-dimensional 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 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.
就有欧氏范数。在这个http://mathworld.wolfram.com/Norm.html
The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. Norms exist for complex numbers (the complex modulus, sometimes also called the complex norm or simply "the norm"), Gaussian integers (the same as the complex modulus, but sometimes unfortunately instead defined to be the absolute square), quaternions (quaternion norm), vectors (vector norms), and matrices (matrix norms). A generalization of the absolute value known as the p-adic norm is also defined.
Norms are variously denoted 
, 
, 
, or 
. In this work, single bars are used to denote the complex modulus, quaternion norm, p-adic norms, and vector norms, while the double bar is reserved formatrix norms.
, , , or . In this work, single bars are used to denote the complex modulus, quaternion norm, p-adic norms, and vector norms, while the double bar is reserved formatrix norms.
The term "norm" is often used without additional qualification to refer to a particular type of norm (such as a matrix norm or vector norm). Most commonly, the unqualified term "norm" refers to the flavor of vector norm technically known as the L2-norm. This norm is variously denoted 
, 
, or 
, and gives the length of an n-vector 
. It can be computed as
, , or , and gives the length of an n-vector . It can be computed as |
The norm of a complex number, 2-norm of a vector, or 2-norm of a (numeric) matrix is returned by Norm[expr]. Furthermore, the generalized 
-norm of a vector or (numeric) matrix is returned by Norm[expr, p].
Read full article from Norm (mathematics) - Wikipedia, the free encyclopedia-norm of a vector or (numeric) matrix is returned by Norm[expr, p].
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