Corollary Hölder's inequality: $\Vert f \Vert_p=\sup_{\{g\::\:\Vert g\Vert _q=1\}}\int fg$

Question

I want to prove the following corollary of Hölder's inequality:

$f\in L^p \Longrightarrow \Vert f \Vert_p=\sup_{\{g\::\:\Vert g\Vert _q=1\}}\int fg$

where $\frac{1}{p}+\frac{1}{q}=1$. I have already proved that $\Vert f \Vert_p \geq \sup_{\{g\::\:\Vert g\Vert _q=1\}}\int fg$, using Hölder's inequality when $\Vert g\Vert _q=1$ and then taking the supreme. However, I am having a lot of trouble proving the other inequality. Can someone help me?

Answer 1

Pick $g = sign(f) \times \frac{1}{\|f^{p-1}\|_q} |f|^{p-1}$ where $sign(f)(x) = 1$ if $f(x) > 0$ and $sign(f)(x) = -1$ if $f(x)\leq 0$.

It is straighforward that $\|g\|_q = 1$. Let us now simplify $\int f g$. First, note that $\|f^{p-1}\|_q = \|f\|_p^{p-1}$, and observe that $sign(f) \times f = |f|$. Thus

\begin{align} \int f g = \frac{1}{\|f\|_p^{p-1}} \int |f|^{p} = \|f\|_p \end{align}

Answer 2

I suppose your functions are real valued and your integrals are w.r.t. Lebesgue measure.

Hint: Let $g_n(x)=\frac 1 {a_n} f(x)\chi_{0<|f(x)| \leq n} \chi_{|x| \leq n}$ where $a_n$ is the $L^{q}$ norm of $f\chi_{0<|f| \leq n} \chi_{|x| \leq n}$. Then the supremum on the right side is at least equal to $\int fg_n$. Compute this integral and let $n \to \infty$.