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where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
The absolute value of a number may be thought of as its distance from zero. In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and . For example, the absolute value of 3 is ...
The absolute value is a norm for the real line; as required, the absolute value satisfies the triangle inequality for any real numbers u and v. If u and v have the same sign or either of them is zero, then | + | = | | + | |. If u and v have opposite signs, then without loss of generality assume | | > | |.
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. In particular, the Euclidean distance in a Euclidean space is defined by a ...
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than (<) and greater than (>).
The absolute difference of two real numbers and is given by , the absolute value of their difference. It describes the distance on the real line between the points corresponding to and . It is a special case of the L p distance for all and is the standard metric used for both the set of rational numbers and their completion, the set of real ...
Landau's inequality provides an upper bound for the absolute values of the product of the roots that have an absolute value greater than one. This inequality, discovered in 1905 by Edmund Landau, [9] has been forgotten and rediscovered at least three times during the 20th century. [10] [11] [12]
The absolute value bars can be dropped when is a rational number with an even numerator in its reduced form, and is drawn from the set of real numbers, or one of its subsets. The Euclidean norm from above falls into this class and is the 2 {\displaystyle 2} -norm, and the 1 {\displaystyle 1} -norm is the norm that corresponds to the rectilinear ...