The space of absolutely continuous functions on $[0,1]$ is equipped with norm $\|f\|:=|f(0)|+V_0^1(f)$.
Every AC-function is continuous with bounded variation, so it is a subspace of $C[0,1]\cap BV[0,1]$.
Is this subspace closed?
2026-02-24 00:27:23.1771892843
absolutely continuous functions as a subspace of continuous functions with bounded variation
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According to a positive answer to an other your question, this space of absolutely continuous functions is complete, so it is closed in a metric space $C[0,1]\cap BV[0,1]$, containing it.