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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.
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] A norm is a generalization of the intuitive notion of "length" in the physical world.
In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().
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 .
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Restricting this extended norm to the bounded functions (i.e., the functions with finite above extended norm) yields a (finite-valued) norm, called the uniform norm on . Note that the definition of uniform norm does not rely on any additional structure on the set X {\displaystyle X} , although in practice X {\displaystyle X} is often at least a ...
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension.It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring.