# Given a sequence of Lp functions, does the integral commute with the lp norm?

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. :)

• Have you considered the fact that on $\mathbb{R}^\infty$, $\|x\|_p=(\sum_{i}^\infty |x|^p)^{1/p}$ is a norm? – Alex R. Mar 7 at 21:59
• Do you mean something like rewriting the inequality as, $|| \{ \int_E f_n d \mu \} ||_p \leq \int_E || \{ f_n(x) \} ||_p d \mu$? – user38770 Mar 7 at 22:11

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.

• Thank you very much for your insightful answer. – user38770 Mar 8 at 4:39