Let $S$ be an infinite set and let $\lbrace s_n \rbrace_{n=1}^{\infty}$ be a sequence of distinct points in $S$. Let $X$ be a Banach space of bounded functions on $S$, supplied with thr supremum norm.
If $x=(x_1, x_2,...) \in \ell^1$, then the definition $\zeta_x (f)= \sum_{n=1}^{\infty} x_n f(s_n)$ yields a bounded linear functional on $X$.
Show that the map $x \to \zeta_x$ is an isometric embedding of $\ell^1$ into $X'$.
I have shown that $\zeta_x (f)$ is a bounded linear functional, how about the map above? how to show it is an isometric embedding?
Although I still have my doubts regarding what was actually intended, the OP seems to have confirmed that he meant the question exactly as stated.
In that case this is trivially false. For example, let $X$ be the space of constant functions on $S$. If $x\in\ell^1$, $x\ne0$ but $\sum x_j=0$ then $\zeta_x=0$.