EDB — 144

[144] Let \(I=[a,b]\). Let \(V\) be the set of functions \(f:[a,b]→ℝ\) that are piecewise constant; it is the vector space generated by \({\mathbb 1}_ J\), all the characteristic functions of all intervals \(J\subseteq I\). Prove that the closure of \(V\) (according to uniform convergence) coincides with the space of regulated functions.

So the space of regulated functions, endowed with the norm \(\| ⋅\| _∞\), is a Banach space.