Space of bounded continuous functions

Remarks

Example: $f(x) = \sin x$ on $X = \R$ is continuous, and $\abs{\sin x} \le 1$ with equality at $x = \pi/2$, so $f \in C_b(\R)$ with $\norm{f}_\infty = 1$.

Non-example: $g(x) = x$ on $\R$ is continuous, but $\sup_x \abs{g(x)} = \infty$, so $g \notin C_b(\R)$.

The supremum is finite by the definition, and the norm axioms follow from those of $\abs{\cdot}$. The space $C_b(X)$ is complete, so it is a Banach space. If $X$ is compact, $C_b(X)$ is the space $C(X)$ of all continuous functions on $X$. If $X$ is a set with the discrete topology, $C_b(X)$ is the space $\ell^\infty(X)$ of bounded functions.