Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set
Find the adjoint of $T$.
I know how to this in the case $p=2$ as shown here. But in general $L^p$ is not an Hilbert space and the definition of adjoint is different: $$\langle Tx,y^*\rangle =\langle x,T^*y^*\rangle $$ with $T:X\to Y$, $y^*\in Y^*$ and $T^*:Y^*\to X^*$.
In this case I do not necessarily have an integral representation of the duality and therefore I do not know how to compute the adjoint.