I think the solution to this question somehow involves Riesz Representation Theorem, but I don't see how to apply it.
Suppose $\{X,\mathcal{M},\mu\}$ is a $\sigma-$ finite measure space, $1\leq p<\infty$ and $\phi$ in a continuous linear functional on $L^{p}(\mu)$.
a) Prove that if $1<p$ then there is an element $f\in L^{p}(\mu)$ such that $\mid\mid f\mid\mid_{p}=1$ and $\phi(f)=\mid\mid\phi\mid\mid$. b) prove that a) need not be true when p=1
For $1 < p < \infty$: If $\phi$ is a continuous linear functional on $L^{p}(\mu)$, then there exists $g \in L^{q}(\mu)$ such that $$ \phi(f)=\int fg\,d\mu,\;\;\; \left[\int |g|^{q}\,d\mu\right]^{1/q} =\|\phi\|. $$ See if you can figure out how to choose $f \in L^{p}$ to make the first integral look like the second, knowing that $1/p+1/q=1$, or $q/p+1=q$. See where it leads.