Can I flip $ \left\lVert a - b \right\rVert_2 \ge \left\lVert a \right\rVert_2 - \left\lVert b \right\rVert_2$ to $\left\lVert a - b \right\rVert_2 \le \left\lVert b \right\rVert_2 - \left\lVert a \right\rVert_2$ using the argument of 2-norm values are positive?
2026-04-07 16:16:37.1775578597
Can I flip $ \left\lVert a - b \right\rVert_2 \ge \left\lVert a \right\rVert_2 - \left\lVert b \right\rVert_2$
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No, you cannot. For instance, if $a\neq0$ and $b=0$, you get$$\|a\|_2=\|a-b\|_2\leqslant\|b\|_2-\|a\|_2=-\|a\|_2.$$That is, a positive number is smaller than a negative one…