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

p, q>1

are called conjugate indices (or Hölder conjugates) if :

\frac + \frac = 1.

Formally, we also define

q = \infty

as conjugate to

p=1

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

p, q>1

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: *

\frac + \frac = 1,

pq = p + q,

\frac = p - 1,

\frac = q - 1.

References

* Antonevich, A. ''Linear Functional Equations'', Birkhäuser, 1999. . Functional analysis Linear functionals {{mathanalysis-stub

Properties

See also

References