# Tag Archives: Metric Space

## Completeness of the bounded-real-function-space B(S)

Let $$S$$ be a nonempty set and $$B(S)$$ a metric space of all the bounded real functions on $$S$$ with the supremum norm $$\Vert f \Vert = \sup \left\{ | f(s) | ~|~ s\in S \right\}$$ and the metric function $$d(f,~g) = \Vert f-g \Vert .$$ Then $$B(S)$$ becomes a complete metric space. To prove… Read More »