Let $L_{k}^{1}([0,1])$ be the space of all $f\in C^{k-1}([0,1])$ such that $f^{(k-1)}$ is absolutely continuous on $[0,1]$ (and hence $f^{(k)}$ exists a.e. and is in $L^{1}([0,1])$). Show that $||f||=\sum_{j=0}^{k}\int_{0}^{1}|f^{(j)}(x)|\;dx$ is a norm on $L_{k}^{1}([0,1])$ that makes it into a Banach space.
I can show that the given expression is a norm but I'm having some trouble showing that the space is complete.
The completeness of this space can be shown by the following argument: Take a cauchy-sequence $f_n$ and consider the sequence of functions $(f_n, f^{(1)}_n, \ldots, f^{(k)}_n) \in L^1([0,1])^{k+1}$. What can be said about this sequence? What follows for the original sequence $f_n$?
Let my know if this helps or if you need additional hints.