Conjugate Indices 1/p + 1/q = 1

Conjugate Indices 1/p + 1/q = 1

 Definition: Let $1 \leq p \leq \infty$. Define the Conjugate Index of $p$ is the number $q$ defined as follows: a) If $1 < p < \infty$ then the $q$ is such that $1 < q < \infty$ such that $\displaystyle{\frac{1}{p} + \frac{1}{q} = 1}$. b) If $p = 1$ then $q = \infty$. c) If $p = \infty$ then $q = 1$.

Note that if $1 < p < \infty$ then we can give an explicit formula for the conjugate index $q$ of $p$. Namely:

(1)
\begin{align} \quad q = \frac{p}{p-1} \end{align}

Since:

(2)
\begin{align} \quad \frac{1}{p} + \frac{1}{q} = \frac{1}{p} + \frac{p-1}{p} = \frac{1 + p - 1}{p} = 1 \end{align}

A few example of $p$ and its conjugate index is given in the table below:

$p$ Conjugate Index, $q$
$1$ $\infty$
$2$ $2$
$3$ $\displaystyle{\frac{3}{2}}$
$4$ $\displaystyle{\frac{4}{3}}$
$5$ $\displaystyle{\frac{5}{4}}$
$\infty$ $1$