Let $(X,||\cdot||_1)$ be a normed space and $Y$ a linear subspace of $X$. Let $||\cdot||_2$ be a norm on $X$ which is equivalent to $||\cdot||_1$ on $Y$. Does there exist a norm on $X$ that coincides with $||\cdot||_2$ on $Y$ and is equivalent to $||\cdot||_1$ on the entire space $X$?
This smells of some version of Hahn-Banach but I dont know where to start. Any clues?