Supremum of integrals of products of functions in $L^p$ space

Question

Here is the problem I'm dealing with

I'm not having success with...well, anything. Any hits on how I could get started and where I would go?

edit: information on $L^p$ space

Answer 1

Three things to use:

1. Triangle inequality for integrals: $\left|\int fg\right| \le \int |fg|$. This yields $\sup_g \left|\int fg\right| \le \sup_g \int |fg|$.
2. Hölder's inequality: $\int |fg|\le \|f\|_p\|g\|_q$. This yields $\sup_g\int |fg|\le \|f\|_p$.
3. Equality case of Hölder's inequality: you need only the easy direction, namely $\int fg= \|f\|_p$ when $g=|f|^{p-2}f /\|f\|_p$. This yields $\|f\|_p\le \sup_g \left|\int fg\right|$, closing the loop.