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

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.

• Did you try something to find the countable base in a direct way? Dec 19, 2020 at 21:05

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.