# functional analysis – Classic excecise of \$L^p\$ spaces but using uniform boundedness principle

I have the following exercise:
Let $$f$$ a real-valued Lebesgue measurable function on $$(0,1 )$$, if $$fg$$ is Lebesgue integrable on $$(0,1)$$ for any function $$gin L^p((0,1),mathbb{R})$$ with $$p in (1,infty)$$, then $$fin L^{frac{p}{p-1}}((0,1),mathbb{R})$$.

This is an exercise I remember from my Lebesgue Measure course, I think we proved the results for simple functions and then extend the result to $$f$$. But here I need to use the uniform boundedness principle, so I think I somehow need a sequence of operators, but I really don’t know how to do this.

Any hints appreciated!