= Exponential of a bounded skew-adjoint operator is unitary
{title2=$T^*=-T\implies(e^T)^*=(e^T)^{-1}=e^{-T}$}
For a bounded <skew-adjoint operator> on a complex <Hilbert space>, the <operator exponential> converges in <norm> because its term <norms> are bounded by $\|T\|^n/n!$. Continuity of the <adjoint operator> gives $(e^T)^*=e^{T^*}=e^{-T}$. <Absolute convergence> permits the Cauchy product of $e^T$ and $e^{-T}$: its degree-$n$ coefficient is $T^n\sum_{j=0}^n(-1)^j/[j!(n-j)!]$, zero for $n>0$ and the identity for $n=0$. Both multiplication orders therefore give the identity, proving unitarity. The boundedness hypothesis makes this elementary power-<series> proof valid without domain issues for unbounded generators.
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