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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.

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

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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.

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