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Problem: For bounded self-adjoint operator $A,B$ on a same Hilbert space $\mathcal{H}$, prove $\left\|e^{iA} - e^{iB}\right\| \leq \|A-B\|$.

This problem has a hint: use mean value theorem to the function $t \to \left\langle e^{-itB}e^{itA}x,y\right\rangle$ on $[0,1]$ for $x,y$ in $\mathcal{H}$. When I use mean value theorem, I got $$\left\langle e^{-iB}e^{iA}x,y\right\rangle - \langle x,y\rangle = \left\langle i(A-B)e^{-it_0B}e^{it_0A}x,y\right\rangle.$$ Then I have no idea to go further, thanks for any help.

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That's not right, but you can write $$\dfrac{d}{dt} \langle e^{-itB} e^{itA} x, y \rangle = \langle (-i e^{-itB} B e^{itA} + i e^{-itB} A e^{itA}) x, y\rangle = \langle i e^{-itB} (A-B) e^{itA} x, y \rangle $$ which is different from what you wrote if $A$ and $B$ don't commute. Next, estimate the absolute value.

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  • $\begingroup$ I already got it $\endgroup$ –  apple Dec 12 '18 at 3:13

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