Consider $\Bbb R^n$ with the standard inner product and let $P$ be an orthogonal projection defined on $\Bbb R^n$. It is known that the operator norm of $P$ induced by the inner product is less than or equal $1$. Let $\|\cdot\|$ be any arbitrary norm on $\Bbb R^n$. Is it true that the operator norm of $P$ induced by the new norm is less than or equal to $1$?