Conjugate index

The conjugate index is a term from mathematics especially from the functional analysis. A positive real number, which is regarded as an index, another positive number is assigned by an equation. This is called the conjugate index. The term is used especially in connection with the spaces, and the Holder 's inequality.

Definition

A positive real numbers is called conjugate index to a positive real number if

Or equivalently

Applies.

Especially is also the number one conjugated index of.

Application

Especially in the integral calculus, but also in the classical analysis as in the Stochastic occur conjugate pairs of numbers. Usually, the first encounter is with two mutually conjugate numbers instead of in the definition of Hölder 's inequality, where the norm of a product of elements by the product of the associated p - and q- norms of the respective elements can be estimated.

Example

The typical example of mutually conjugate numbers is the number 2, which is conjugated to itself. In most cases, the special cases of statements about conjugated numbers are mainly of historical interest, such as the above-mentioned Hölder 's inequality is a later generalization of the Cauchy- Schwarz inequality.