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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.
A norm induces a distance, called its (norm) induced metric, by the formula (,) = ‖ ‖. which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space .
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 .
The Frobenius norm defined by ‖ ‖ = = = | | = = = {,} is self-dual, i.e., its dual norm is ‖ ‖ ′ = ‖ ‖.. The spectral norm, a special case of the induced norm when =, is defined by the maximum singular values of a matrix, that is, ‖ ‖ = (), has the nuclear norm as its dual norm, which is defined by ‖ ‖ ′ = (), for any matrix where () denote the singular values ...
Every inner product space induces a norm, called its canonical norm, that is defined by ‖ ‖ = , . With this norm, every inner product space becomes a normed vector space. So, every general property of normed vector spaces applies to inner product spaces.
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).
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.