Closed subset in different norms

Question

Let us say I have a space of functions $X$ (e.g. $X=C([0,1],[0,1])$ continuous from the unit interval into itself) and two subspaces $S_1$ and $S_2$ such that $S_2\subset S_1(\subset X)$ (e.g. $S_i=L^i([0,1],[0,1])$ for $i\in\{1,2\}$). Suppose that $S_i$ is a Banach space under norm $\|\cdot\|_i$ for $i\in\{1,2\}$ (e.g.$\|\cdot\|_i = \|\cdot\|_{L^i}$). Suppose that I give you any closed subset $A$ of $S_2$ (with the $\|\cdot\|_2$ norm). Is this subset $A$ also closed in $S_1$ (with the $\|\cdot\|_1$ norm)? My intuition tells me it is not, but I would appreciate a counterexample.