Let $X = AC[0,1]$ be the space of all absolutely continuous functions from $[0, 1]$ to $\mathbb{R}^n$, and let $X'=\text{The space of all Lebesgue integrable functions}$.
Consider the linear function $F: X \to X'$ defined by $F(f)=f'$. In order to $F$ be well-defined we only consider $f'$ where it exists (It almost every where exists on $[0,1]$). Clearly $F$ is linear, my question is that, is there any interesting norm on $X$ or a subspace $Y \subset X$ with a suitable norm such that it turns $F$ a continuous function ?
I'm not looking for trivial norms. This operator frequently appear in optimal control, and the motivation of this question is here
In one dimension, you have $AC[0,1] = W^{1,1}(0,1)$, which is a Sobolev space. Moreover, your space $X'$ coincides with $L^1(0,1)$. Thus,
$$\| F(f) \|_{L^1} = \| f' \|_{L^1} \le \| f \|_{W^{1,1}}.$$ Therefore, $F$ is bounded with these natural norms.