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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?

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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 $U$.

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