Is the following equation correct? How to prove.
$$ \sup_{\|x\|=1}\|Ax\|=\sup \frac{\|Ax\|}{\|x\|} $$
Thanks!
2026-03-28 02:22:08.1774664528
How to prove two suprema are equal
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For any $x \neq 0$ the vector $y=\frac 1 {\|x||} x$ has norm $1$, and $\frac {\|Ax\|} {\|x\|}=\|Ay\|$ by linearity. So RHS $\leq $ LHS. The other inequality is obvious.