The space of absolutely continuous functions is separable with norm $\|F\|_{AC}=\sup |F|+\int_a^b|F'|$

Question

Consider the space of absolutely continuous functions $AC([a,b])$ equipped with norm $$ \|F\|_{AC}=\sup |F|+\int_a^b|F'| $$

It can be shown that $AC([a,b])$ is a Banach space under this norm.

Please try to prove that $AC([a,b])$ is separable, i.e. there is a countable dense subset.

Answer 1

One can prove directly that it is separable, which is always a bit of work. A more powerful (but elegant) way to prove it may be the following:

Note that, using a decomposition of absolutely continuous functions, $AC([a,b])$ is isometrically isomorphic to the direct sum of $L^1([a,b])$ and a one-dimensional space. Since separability is stable under isometric isomorphisms and finite direct sums, the result follows from the separability of $L^1([a,b])$ and the one-dimensional space.