Consider the Banach space $X=C[0,1]$ of real continuous function on $[0,1]$ equipped with the supremum norm. Consider the operator $A:D(A)\to X$, $Af=f'$ for each $f\in D(A)=C^1[0,1]$. We can see that the domain of $A$ is dense in $X$. But we know that $A$ is unbounded (non continuous) on its domain. If we take $Y=$ The space of constant functions, then $A$ is a bounded operator on this subspace.
Do we have the existence of a maximal subspace $Y\subset X$ such that $A:Y\to X$ is a bounded operator ?