> [!definition] > > Let $X$ be a [[Set|set]]. The space $B(X, \real)$ and $B(X, \complex)$ ($B(X)$) are the spaces of bounded real-valued and complex-valued [[Function|functions]], respectively. > [!theorem] > > Let $X$ be a set, then $B(X)$ is [[Complete Metric Space|complete]] with respect to the [[Uniform Norm|uniform norm]]. > > *Proof*. Every [[Cauchy Sequence|Cauchy sequence]] is bounded. Evaluating the limit pointwise yields another bounded function.