# An application of Holder's Inequality

Suppose $$1\leq p,q\leq \infty$$ and $$1/p+1/q=1$$. Let $$f\in\mathcal{L}^p(E)$$. Show that $$f=0$$ a.e. if and only if \begin{align*} \int_E f\cdot gdm=0 \end{align*} for all $$g\in \mathcal{L}^q(E)$$. Hint: Choose a smart $$g$$ so that $$\int_Ef\cdot gdm=\|f\|_p^p$$.

• In fact, if $f \in L^p(E)$ then $$\|f\|_p = \sup_{g \in L^q(E)} \int_E fg \, dm.$$ – Umberto P. Dec 4 '18 at 20:20

Hint: try $$g=\mathrm{sign}(f)\cdot|f|^\alpha$$ for some $$\alpha\in\mathbb R$$.
Second hint: $$\alpha=p/q$$.