Let $O$ be a bounded open set in $\mathbb{R}^n$. And let $1<p<\infty$.
Then I know that each $L^p(O)$ is uniformly convex and has a schauder basis.
Now, the duality map is then uniquely defined for each element of $L^p(O)$.
However, I cannot see why the image of a Schauder basis in $L^p(O)$ by the duality map becomes a Schauder basis in $L^q(O)$ where $1/p+1/q=1$. I tried everything I can think of but the duality map is not linear...So things are just getting more twisted.
Could anyone please explain to me?