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In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:}. where denotes the supremum.
The norm of a vector is represented with double bars on both sides of the vector. The norm of a vector v can be represented as: ‖ ‖ The norm is also sometimes represented with single bars, like | |, but this can be confused with absolute value (which is a type of norm).
Hierarchy of mathematical spaces. Normed vector spaces are a superset of inner product spaces and a subset of metric spaces, which in turn is a subset of topological spaces. In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. [1]
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm.Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. [8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.
The perimeter of the square is the set of points in ℝ 2 where the sup norm equals a fixed positive constant. For example, points (2, 0), (2, 1), and (2, 2) lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2.
By Dvoretzky's theorem, every finite-dimensional normed vector space has a high-dimensional subspace on which the norm is approximately Euclidean; the Euclidean norm is the only norm with this property. [24] It can be extended to infinite-dimensional vector spaces as the L 2 norm or L 2 distance. [25]