Existence and uniqueneess of functional maximizing value on specific element of $L^p$

Question

Given a function $f\in L^p$ with $1<p<\infty$ and $\|f\|_p=1$ is it true that there is a unique function $g \in L^q$ with $\|g\|_q=1$ such that $\int fg =1$? This is over an arbitrary measure space.

Answer 1

Yes, because $L^q$, which is the dual of $L^p$, is strictly convex for $1<q<\infty$ (actually, it’s uniformly convex). Then you can conclude by the statement contained in this post.

P.s. Moreover, $g=\operatorname{sgn}(f)|f|^{p-1}$ a.e. (you can verify that the $L^q$ norm is $1$ and $\int fg=1$).