Conjugated Index

The conjugated index is a term from mathematics , especially from functional analysis . A positive real number , which is understood as an index, is assigned another positive number by an equation and is called a conjugate index. The term is used especially in connection with the spaces and the Hölder inequality . ${\ displaystyle L ^ {p}}$

definition

A positive real number is called the conjugate index to the positive real number , if ${\ displaystyle q}$ ${\ displaystyle p}$

{\ displaystyle {\ begin {aligned} {\ frac {1} {p}} + {\ frac {1} {q}} & = 1 \\\ Leftrightarrow p + q & = p \ cdot q \\\ Leftrightarrow 1 & = (p-1) \ cdot (q-1) \ end {aligned}}}

applies. In particular, the number is then also a conjugate index of . ${\ displaystyle p}$ ${\ displaystyle q}$

application

Especially in integral calculus , but also in classical analysis and stochastics , conjugate pairs of numbers appear. Usually the first encounter with two conjugated numbers takes place in the definition of the Hölder inequality , where the norm of a product of elements can be estimated by the product of the p- and q-norms of the respective elements.

example

The typical example of numbers conjugated to one another is the number 2, which is conjugated to itself. The special cases of statements about conjugate numbers with are mostly historically interesting, for example the Hölder inequality mentioned above is a later generalization of the Cauchy-Schwarz inequality . ${\ displaystyle p = q = 2}$

Web links

conjugate index on PlanetMath

Individual evidence

^ Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson: Real analysis . Prentice-Hall, Upper Saddle River, NJ 1997, ISBN 978-0-13-458886-5 , pp. 536 .

[1] Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson: Real analysis . Prentice-Hall, Upper Saddle River, NJ 1997, ISBN 978-0-13-458886-5 , pp. 536 .