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In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition [ edit ]
Normal map (a) is baked from 78,642 triangle model (b) onto 768 triangle model (c). This results in a render of the 768 triangle model, (d). In 3D computer graphics , normal mapping , or Dot3 bump mapping , is a texture mapping technique used for faking the lighting of bumps and dents – an implementation of bump mapping .
Every (real or complex) vector space admits a norm: If = is a Hamel basis for a vector space then the real-valued map that sends = (where all but finitely many of the scalars are ) to | | is a norm on . [9] There are also a large number of norms that exhibit additional properties that make them useful for specific problems.
Every real -by-matrix corresponds to a linear map from to . Each pair of the plethora of (vector) norms applicable to real vector spaces induces an operator norm for all -by-matrices of real numbers; these induced norms form a subset of matrix norms.
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.
Norm map, a map from a pointset into the ordinals inducing a prewellordering Norm group , a group in class field theory that is the image of the multiplicative group of a field Norm function, a term in the study of Euclidean domains , sometimes used in place of "Euclidean function"
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 .
More generally, for each real the map : defined by (, …,):= (= ()) is a semi norm. For each p {\displaystyle p} this defines the same topological space. A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space ...