Supremum Metric

Definition

Consider a set $X$, a metric space $(M,d^{\prime })$, and the set of all bounded continuous functions $C(X,M)$ on $X$ to $M$. Then, the supremum metric $d$ on $X$ is defined as $d(f,g)=\sup \{d^{\prime }(f(x),g(x))|x\in X\},\forall f,g\in C(X,M)$