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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.
The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules =, =, = =.In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
In number theory norm forms are studied as Diophantine equations, where they generalize, for example, the Pell equation. [2] For this application the field K is usually the rational number field, the field L is an algebraic number field , and the basis is taken of some order in the ring of integers O L of L .
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 with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.
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 .
Asymmetric norms differ from norms in that they need not satisfy the equality () = (). If the condition of positive definiteness is omitted, then p {\displaystyle p} is an asymmetric seminorm . A weaker condition than positive definiteness is non-degeneracy : that for x ≠ 0 , {\displaystyle x\neq 0,} at least one of the two numbers p ( x ...
y or x · y. Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol, [6] and it must be explicitly used; for example, 3x is written as 3 * x. Rather than using the ambiguous division sign (÷), [a] division is usually represented with a vinculum, a horizontal line, as in 3 / x ...
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). [4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with ...
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