If $X$ is a discrete metric space, then $l^{\infty}(X)$ is isomorphic to $C(\beta X)$ where $\beta X$ is the Stone-Cech compactification. Given an $f \in l^{\infty}(X)$, f is continuous because $X$ is discrete, then there is a unique extention $g \in C(\beta X)$. Is it the isomorphism?