Bi-Lipschitz maps and identity

functional-analysis projection metric-spaces

If $k$ is the Lipschitz constant of $L$, then for any $x$, $\|L(x)\|\leqslant k\|x\|$. The triangle inequality thus yields $$ \|x-L(x)\| \leqslant \|x\|+ \|L(x)\| \leqslant (1+k)\|x\| $$

This bound is sharp since for $L(x) = -x$, it holds that $$ \|x-L(x)\| = 2\|x\| $$

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