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? 



Extension of equivalent norms
Yes. Suppose $\|y\|\_2 \le A \|y\|\_1$ for $y \in Y$. Let $U$ be the convex hull of $1/A$ times the unit ball of $X$ and the $\|\cdot \|\_2$-unit ball of $Y$. Then $U \cap Y$ is the $\|\cdot \|\_2$-unit ball of $Y$. Use the gauge functional of $U$.


