I was wondering if the spectral norm is a Lipschitz function with respect to the spectral norm. How can we prove whether it is or not? In other words, is $$\big| \|X\| - \|Y\| \big| \le L \|X-Y\|$$ for some $L$?
2026-03-25 17:37:41.1774460261
Is the spectral norm a Lipschitz function with respect to the spectral norm?
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Any norm is Lipschitz of rank 1 with respect to itself. This follows from the traingle inequality.
Since $\|X\| \le \|Y\|+\|X-Y\|$ we have $\|X\| -\|Y\| \le \|X-Y\|$. Reversing the roles of $X,Y$ gives $\|Y\| -\|X\| \le \|X-Y\|$ and combining these gives $| \|X\|-\|Y\| | \le \|X-Y\|$.