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 numbers
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; 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
References
* Antonevich, A. ''Linear Functional Equations'', Birkhäuser, 1999. .
Functional analysis
Linear functionals
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