I'm hinted by previous posts that this proof requires the use of the compactness theorem but I don't know how. Could somebody outline such a proof? Again, I am trying to prove that $x^* \neq 0$ such that $\| Ax^*\| = \| A\| \| x^*\|$.
2026-04-08 23:06:38.1775689598
Show there's an $x^* \neq 0$ such that $\| Ax^*\| = \| A\| \| x^*\|$
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Note that it is enough to prove it for those $x $ with $\|x\|=1$. Then use that the unit sphere, in finite dimension, is compact.