The following pictures are Theorem 2.14 (The dual of $L^p(\Omega)$ in Lieb's Analysis book and its proof of the case $1<p<\infty$.
My question is how to get the inequility (3) in the red box? It really makes me wonder...
Edit: Add the picture of Lemma 2.8 at the bottom.
Unless I've missed something, the authors made a little mistake there, but the result is of course true. Lemma 2.18 states that there exists $h\in K$ such that $$ \operatorname{Re}\int (k-h)u \leq 0 $$ for all $k\in K$. This can be rewritten into $$ \operatorname{Re}\int ku \leq \operatorname{Re}\int hu $$ for all $k\in K$. Not having $0$ on the right hand side is actually not a problem here, because $K$ is indeed a complex linear space, and thus $$ \lambda\operatorname{Re}\int ku \leq \operatorname{Re}\int hu\quad\text{and}\quad \lambda\operatorname{Im}\int ku \leq \operatorname{Re}\int hu $$ for all $\lambda \in \mathbb R$, thus $\int ku=0$.
I checked their errata, but this does not seem to be in there. If you feel like this is important, and you're confident it is a mistake, you can send them an email about it.