Hölder Conjugate

In mathematics, 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, 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:
* $\frac + \frac = 1,$
* $pq = p + q,$
* $\frac = p - 1,$
* $\frac = q - 1.$

See also

* Beatty's theorem

References

* Antonevich, A. ''Linear Functional Equations'', Birkhäuser, 1999. .

Functional analysis
Linear functionals

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