> [!definition] > > Let $X$ be a set and $f \in B(X)$ be a [[Space of Bounded Functions|bounded function]], define the **uniform norm** of $f$ as > $ > \norm{f} = \sup\bracs{|f(x)|: x \in X} > $ > then $\norm{\cdot}$ induces a [[Metric Space|metric]], and [[Limit|convergence]] with respect to this norm corresponds to [[Uniform Convergence|uniform convergence]].