Let $E$ be a measurable set, $1\leq p<\infty$, and $q$ the conjugate of $p$ (i.e., $\frac{1}{p}+\frac{1}{q}=1$). Suppose $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g\in L^q(E)$, the sequence $\{\int_Eg\cdot f_n\}$ is bounded. Show that $\{f_n\}$ is bounded in $L^p(E)$.
This problem is from Royden's real analysis book (Chapter 8, Ex 26). It is required that the proof cannot directly make use of the Uniform Boundedness Principle.