The author makes the following claim, which I have generalized to more abstract setting:
Given a compact operator $U:X^{\ast}\rightarrow l_{1}$, we know that $\sup_{\varphi\in B_{X^{\ast}}}\|U\varphi\|_{l^{1}}<\infty$, but why it follows that $\displaystyle\sum_{n=1}^{\infty}|(U\varphi_{n})_{n}|<\infty$ for every sequence $\{\varphi_{n}\}$ in $B_{X^{\ast}}$?
I use the notation $U\varphi$ as an element in $l_{1}$ by writing its componentwise as $U\varphi=(U\varphi)_{n}$.
Here is another related question. I manage to use the totally boundedness of $U(B_{X^{\ast}})$ to show the existence of some positive integer $N$ such that $\sup_{\varphi\in B_{X^{\ast}}}\displaystyle\sum_{n=N}^{\infty}|(U\varphi)_{n}|<\epsilon$, where $\epsilon>0$ is given, is there any other method to choose such an $N$?