I have a measurable set $E$ and a sequence of functions $f_n$ in $L^p(E)$ such that for all functions $g\in L^q(E)$, with $q$ the conjugate index of $p$, $\displaystyle\int_{E}f_ng$ is bounded. I have to show that the sequence $f_n$ itself is bounded in $L^p(E)$.
This problem appears in Royden's analysis book and I get that I need to apply the uniform boundedness principle. I've started out by considering the family ${f_n}$ as a family of operators from $L^q\rightarrow\overline{\mathbb R}$ taking $g\in L^q$ to $\displaystyle\int_{E}f_ng$. I know that this is bounded for all $g$ in my domain. However, I don't get how to apply the result of the uniform boundedness principle after this. The principle itself allows me to conclude that $\displaystyle \sup_{n,\lVert g\rVert_q=1}\displaystyle\int_{E}gf_n<\infty$. How do I get to $\lVert f_n\rVert_p$ is bounded from here? I think I'm making some mistake. I tried Holder's inequality, but I don't seem to be able to finish the proof.