I'm working on (a), (b) and (c) of this exercise.
To (a): T.p.: every Cauchy sequence in E has a convergent subsequence with limit in E.
So, take $(x_n)_n \in E $. $(x_n)_n$ is a sequence of sequences: $(x_n)_n = ((x_{1n})_n,(x_{2n})_n,(x_{3n})_n,...)$, where $(x_{in})_n \in E_i \forall i \in \mathbb{N}$.
Let now $(x_n)_n \in E $ be cauchy, then I get for $\epsilon > 0$ arbitrary and n, m sufficiently large: $\epsilon > \lVert ((x_{1n})_n,(x_{2n})_n,(x_{3n})_n,...) - ((x_{1m})_m,(x_{2m})_m,(x_{3m})_m,...) \rVert = (\sum \limits_{i=1}^\infty (\lVert x_{in} - x_{im} \rVert)^p)^\frac{1}{p} > \lVert x_{in} - x_{im} \rVert$, so $x_{in}$ is cauchy in $E_i$ with limit $y_i$. Now I wanted to show that $(y_n)_n$ is the limit of $(x_n)_n$, but I'm not able to do so, because the sum is not going to $N$, but to $\infty$.
How can I do this?
And could you please help me with (b) and (c)? I'm struggling how to start there..
For (a):
Let $\{\{S(n)_{j,k}\}_j\}_k$ be the partial sequence of $\{\{x_{j,k}\}_j\}_k$; that is, for all $k$, $$S(n)_{\cdot,k}\in\sideset{}{_p}\bigoplus_{j=1}^n{E_j}$$ with, for all $j<n$, $S(n)_{j,k}=x_{j,k}$. You have shown that, for any $n$, $S(n)_{j,k}\to S_{j,\infty}$.
But the rest of the sequence is small! To be more precise: because $\sum_{j=1}^{\infty}{x_{j,k}^p}<\infty$, for any $\epsilon$, $\sum_{j=J(\epsilon)}^{\infty}{x_{j,k}^p}<\epsilon$. We will abuse notation and write this last inequality as $|\{x_{j,k}\}_j-\{S(J(k,\epsilon))_{j,k}\}_j|<\epsilon^p$ for some $J(k,\epsilon)$.
The final ingredient necessary is to note that although our quantifiers form a loop (we need $J$ to pick $K$ and vice-versa), we can elide the problem by approximating $\{x_{j,k}\}_j$ with a nearby $\{x_{j,\kappa}\}_j$ by Cauchyness.
So, fix $\epsilon$. There exists $K_1\in\mathbb{Z}^+$ such that, for all $k\geq K_1$, $|\{x_{j,K_1}\}_j-\{x_{j,k}\}_k|_p<\frac{\epsilon}{4}$. There exists $N_1\in\mathbb{Z}^+$ such that, for any $n\geq N_1$, $|\{x_{j,K_1}\}_j-\{S(n)_{j,K_1}\}_j|<\frac{\epsilon}{4}$. And there exists $N_2\in\mathbb{Z}^+$ such that, for any $n\geq N_2$, $|\{x_{j,\infty}\}_j-\{S(n)_{j,\infty}\}_j|<\frac{\epsilon}{4}$. Finally, let $N=\max{(N_1,N_2)}$. Then there exists $K_2$ such that, for any $k\geq K_2$, $|\{S(n)_{j,k}\}_j-\{S(n)_{j,\infty}\}_j|_p<\frac{\epsilon}{4}$.
Thus, for any $k\geq\max{(K_1,K_2)}$, \begin{align*} |\{x_{j,k}\}_j-\{x_{j,\infty}\}_j|_p&=|\{S(N)_{j,k}\}_j-\{x_{S(N),\infty}\}_j|_p+{}\\ &\phantom{= |}|(\{x_{j,k}\}_j-\{S(N)_{j,k}\}_j)-(\{x_{j,\infty}\}_j-\{S(N)_{j,\infty}\}_j)|_p \\ &<\frac{\epsilon}{4}+|(\{x_{j,k}\}_j-\{S(N)_{j,k}\}_j)-(\{x_{j,\infty}\}_j-\{S(N)_{j,\infty}\}_j)|_p \\ &\leq\frac{\epsilon}{4}+|\{x_{j,k}\}_j-\{S(N)_{j,k}\}_j|_p+|\{x_{j,\infty}\}_j-\{S(N)_{j,\infty}\}_j|_p \\ &<\frac{\epsilon}{4}+|\{x_{j,k}\}_j-\{S(N)_{j,k}\}_j|_p+\frac{\epsilon}{4} \\ &\leq\frac{\epsilon}{2}+|\{x_{j,k}\}_j-\{x_{j,K_1}\}_j|_p+|\{x_{j,K_1}\}_j-\{S(N)_{j,K_1}\}_j|_p \\ &<\frac{\epsilon}{2}+\frac{\epsilon}{4}+\frac{\epsilon}{4}=\epsilon \end{align*}
(You'll notice that this proof is identical to the case $E_j=\mathbb{R}$.)
For (b): We remarked above that (a) was identical to $(\mathbb{R}^n;l^p)$. The same holds here: you should be able to straightforwardly show that, if we interpret an element of $\sideset{}{_q}\bigoplus_{j=1}^{\infty}{E_j'}$ as a functional on $\sideset{}{_p}\bigoplus_{j=1}^{\infty}{E_j}$ with action given by termwise application and then an infinite sum the result is well-defined. Conversely, given a functional in $\left(\sideset{}{_p}\bigoplus_{j=1}^{\infty}{E_j}\right)'$, the restriction, for all $k$ to $E_k\oplus\left(\sideset{}{_p}\bigoplus_{k\neq j=1}^{\infty}{0}\right)$ ought to give you an element of $E_k'$. Furthermore, the action of the original functional is the sum of the functionals so generated. Because $E_k'$ norms $E_k$, a double-application of Hölder's inequality ought to give you (a) that the sum defining $\sideset{}{_q}\bigoplus_{j=1}^{\infty}{E_j'}$ is satisfied by this construction and (b) the the bijection is norm-preserving.
I suppose you also need to prove linearity, but that takes no more than a line.
For (c): note that, by disjointness of $\{\Omega_n\}_n$, any $f\in L^p(\Omega)$ has a unique decomposition $f=\sum_n{\chi_{\Omega_n}f}$, where $\chi_{(\cdot)}$ is the characteristic function. The result should then follow from showing each $\chi_{\Omega_n}f\in L^p(\Omega_n)$ and definition-chasing.