EDB — 144

[144] Let $I=[a,b]$. Let $V$ be the set of functions $f:[a,b]\to ℝ$ 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 $\Vert \cdot {\Vert }_{\infty }$, is a Banach space.