In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, two
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
are called conjugate indices (or Hölder conjugates) if
:
Formally, we also define
as conjugate to
and
vice versa.
Conjugate indices are used in
Hölder's inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces.
The numbers an ...
, as well as
Young's inequality for products
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality.
Young's inequality ...
; the latter can be used to prove the former. If
are conjugate indices, the spaces ''L''
''p'' and ''L''
''q'' are
dual to each other (see
''L''''p'' space).
Properties
The following are equivalent characterizations of Hölder conjugates:
*
*
*
*
See also
*
Beatty's theorem
In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about th ...
References
* Antonevich, A. ''Linear Functional Equations'', Birkhäuser, 1999. .
Functional analysis
Linear functionals
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