Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and $g \in L^{p'}(\Omega).$ Let $\Lambda:L^p(\Omega)\rightarrow \mathbb{R}$ be the linear function $$ \Lambda f=\int_{\Omega}fg \ d \mu.$$
Show that $\Lambda$ is continuous and $\Vert\Lambda\Vert_*=\Vert g \Vert_{p'}.$
Using Hölder's inequality we see that for any $f$: $$\vert\Lambda f\vert\leq \int_{\Omega} \vert fg \vert \ d \mu\leq \Vert f \Vert_p \Vert g \Vert_{p'}.$$
Since $\Lambda$ is linear we can conclude that it's continuous. We can also conclude that $\Vert\Lambda\Vert_*\leq\Vert g \Vert_{p'}.$ What $f$ should I consider to prove the equality? Or is there a better way of proving this?