---- > [!definition] Definition. ([[dual exponent]]) > For $1 \leq p < \infty$, the **dual exponent**, **conjugate exponent**, or **conjugate index** of $p$, denoted by $p'$, is the element of $[1, \infty]$ such that $\frac{1}{p}+\frac{1}{p'}=1.$ > > > > The following are manifestly equivalent characterizations: > - $p'=\frac{p}{p-1}$ > - $p+p'=pp'$ > - $\frac{p}{p'}=p-1$ > - $\frac{p'}{p}=p'-1$ > > > [!basicexample] > - $1'=\infty$, $\infty'=1$, $2'=2$, $4'=\frac{4}{3}$, $\left( \frac{4}{3} \right)'=4$. > ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```