I have been struggling to prove the following:
Let $ \{ f_n \}$ be a sequence in $ L^p(E) $ for some $ p \geq 1 $. Then,
$$ \left( \sum_{n=1}^\infty | \int_E f_n \mathrm{d}\mu |^p \right)^{ \frac{1}{p}} \leq \int_E \left( \sum_{n=1}^\infty |f_n|^p \right)^{\frac{1}{p}} \mathrm{d} \mu $$.
I'm not sure if any of my attempts have been promising enough to include. Any hints would be greatly appreciated. :)
Let $p >1$. We have $\sum |a_n|^{p} =\sup \{ |\sum a_n b_n|: \sum |b_n|^{q} \leq 1\}$ where $q$ is the index conjugate to $p$. Hence it is enough to show that $|\sum b_n \int_E f_n \,d\mu| \leq $ RHS whenever $\sum |b_n|^{q} \leq 1$. This is very easy since $|\sum b_n f_n| \leq (\sum |f_n|^{p})^{1/p} (\sum |b_n|^{q})^{1/q}$. The case $p=1$ is trivial.