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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 entries are sorted according to increasing norm x 2 + y 2 (sequence A001481 in the OEIS). The table is complete up to the maximum norm at the end of the table in the sense that each composite or prime in the first quadrant appears in the second column. Gaussian primes occur only for a subset of norms, detailed in sequence OEIS: A055025.
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. In mathematical analysis , the uniform norm (or sup norm ) assigns, to real- or complex -valued bounded functions f {\displaystyle f} defined on a set S {\displaystyle S} , the non-negative number
Every norm, seminorm, and real linear functional is a sublinear function.The identity function on := is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation . [5] More generally, for any real , the map ,: {is a sublinear function on := and moreover, every sublinear function : is of this ...
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The norm, N L/K (α), is defined as the determinant of this linear transformation. [ 1 ] If L / K is a Galois extension , one may compute the norm of α ∈ L as the product of all the Galois conjugates of α :
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