TITLE: Space of bounded functions is reflexive if the domain is finite
QUESTION [6 upvotes]: Let $C_b(X)$ be a space of bounded continuous functions on a locally compact space $X$ equipped with the supremum norm. How to show that $C_b(X)$ is reflexive if and only if $X$ is finite?

REPLY [5 votes]: If X is infinite, then $C_b(X)$ contains a copy of $c_0$.