Let $V$ be a normed space. Assume that there exists two equivalent norms on $V$ denoted respectively $\|\cdot\|_1$ and $\|\cdot\|_2$.
If $A$ is a closed subspace of $V$ with respect to $\|\cdot\|_1$. It is true that $A$ is also a closed subspace of $V$ with respect to $\|\cdot\|_2$?
Yes: equivalent norms always define the same topology. To see this, try to show that an open $\| \cdot \|_1-$ball centered at $x$ always contains an open $\| \cdot \|_2-$ball centered at $x$ and viceversa. Can you go on from here?