Hölder's inequality different condition for equality ($L^p$ space) I came across this condition for equality for Hölder's of which I would like to know  proof of. Equality holds if and only if $A \vert f \vert^{p-1} = B \vert g \vert$ for non negative $A,B$ not both zero. (I was under the impression that there is only $\vert f \vert^p = C \vert g \vert^q$ for which the proof is clear).
Any help is appreciated

The exponents are conjugate, so $p+q=pq$. If we take the first condition, then $A^q|f|^{p+q-q}=B^q|g|^q$ (after having took the exponent $q$). Take the power $\frac 1q$ to get the opposite direction.
