The problem:
Let $p\in(0,\infty)$ and $(E_n)_{n\in\Bbb N}$ a sequence of Banach Spaces. If we define $$\Bigl(\sum E_n\Bigr)_p:=\biggl\{ (x_n)_{n\in\Bbb N}~|~x_n\in E_n, \forall n\in\Bbb N, ~\text{and}~\|(x_n)_{n\in\Bbb N}\|_p:=\biggl(\sum_{n\in\Bbb N}\|x_n\|_{E_n}^p\biggr)^{\frac{1}{p}}<\infty\biggr\},$$ then $\Bigl(\bigl(\sum E_n\bigr)_p,\|\cdot\|_p\Bigr)$ is a Banach space and $\Bigl(\bigl(\sum E_n\bigr)_p\Bigr)'$ is isometrically isomorphic to $\bigl(\sum E_n'\bigr)_{p'}$, were $p'>1$ is such that $\frac{1}{p}+\frac{1}{p'}=1$.
Attempt: The demonstration that $\Bigl(\bigl(\sum E_n\bigr)_p,\|\cdot\|_p\Bigr)$ is a Banach space is analogous to that of $\ell_p$. Define $$\begin{align} \Phi:\Bigl(\sum E_n'\Bigr)_{p'}&\longrightarrow\biggl(\Bigl(\sum E_n\Bigr)_p\biggr)'\\ f=(f_n)_{n\in\Bbb N}&\longmapsto \Phi(f):\Bigl(\sum E_n\Bigr)_p\to\Bbb R\\ &~~~~~~~~~~~~~~~~~~~~~~~~~(x_n)_{n\in\Bbb N}\mapsto\sum_{n\in\Bbb N}f_n(x_n). \end{align}$$ I can use Hölder to prove that $\Phi$ is well defined, is continuous and, for all $f\in\bigl(\sum E_n'\bigr)_{p'}$, we have $\|\Phi(f)\|\leq\|f\|$, but I am having a hard time trying to prove that $\|\Phi(f)\|\geq\|f\|$. The same goes for showing that $\Phi$ is onto. Can someone help me with hints or something?
