I would like to prove the following:
I am not asking for a solution! I would simply like a bit of guidance, I can't seem to get started on this problem. Would the proof involve using weak convergence? Perhaps there is a useful theorem for this I can't recall off the top of my head. Any help is greatly appreciated.
(b) $\Rightarrow$ (a): you can use Young's inequality to obtain $$ f_n(x_n) \le \|f_n\|_{X^*}\|x_n\|_X \le \frac1q\|f_n\|_{X^*}^q + \frac1p \|x\|_X^p. $$ (a) $\Rightarrow$ (b): Construct a sequence $x_n$ to be suitable multiples of elements $z_n\in X$ that satisfy $f_n(z_n)=\|f_n\|_{X^*}$, $\|z_n\|_X=1$.
Set $x_n=t_nz_n$, $t_n\in \mathbb R$, choose $t_n$ such that $\sum_{n=1}^\infty f_n(x_n)<\infty$, by using the convergence of $\sum_{n=1}^\infty \|f_n\|^q$. Then prove convergence of $\sum_{n=1}^\infty \|x_n\|_X^p$ using that $\frac1p+\frac1q=1$.
No weak convergence, just playing with series and inequalities in $l^p$-norms.
Hope this helps.