Space of bounded continuous functions

Summary

For a topological space $X$, the space $C_b(X)$ consists of the bounded continuous scalar functions on $X$, with the supremum norm. It is a Banach space: see completeness of the space of bounded continuous functions and the example of the sup norm. Convergence in this norm is uniform convergence on $X$. When $X$ is compact, every continuous function is bounded and $C_b(X)$ is the classical space $C(X)$.