Isomorph vector space such that X complete and Y not Let $X,Y$ be some isomorph vector spaces and let $X $ be a Banach space. If this isomorphism is isometric $Y $ is complete, too. Could someone provide an example such that $Y $ is not complete?

Let $X=C\bigl([0,1]\bigr)$ and $Y=\\{f\in F\mid f\text{ is differentiable and }f(0)=0\\}$. Then$$\begin{array}{rccc}D\colon&Y&\longrightarrow&X\\\&f&\mapsto&f'\end{array}$$is a vector space isomorphism. However, $X$ is complete, whereas $Y$ isn't (in both cases, with respect to the $\sup$-norm).