Conjugate index

In mathematics, two real numbers [math]\displaystyle{ p, q\gt 1 }[/math] are called conjugate indices (or Hölder conjugates) if

[math]\displaystyle{ \frac{1}{p} + \frac{1}{q} = 1. }[/math]

Formally, we also define [math]\displaystyle{ q = \infty }[/math] as conjugate to [math]\displaystyle{ p=1 }[/math] 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 [math]\displaystyle{ p, q\gt 1 }[/math] are conjugate indices, the spaces L^p and L^q are dual to each other (see L^p space).

See also

• Beatty's theorem

References

• Antonevich, A. Linear Functional Equations, Birkhäuser, 1999. ISBN:3-7643-2931-9.