= Banach algebra exponential
{c}
{title2=$e^x=\sum_{n=0}^\infty x^n/n!$}
The exponential series converges absolutely in a unital <Banach algebra>. Multiplication of absolutely convergent series gives $e^xe^{-x}=1$, and gives $e^{x+y}=e^xe^y$ when $x,y$ commute. Termwise differentiation gives $e^{tx}=1+tx+O(t^2)$ near zero. In a <C-star algebra>, if $h=h^*$ and $t$ is real, taking adjoints termwise gives $(e^{ith})^*=e^{-ith}$, so this exponential is a <Unitary element of a C-star algebra> and has norm one. These facts let a norm bound on a functional recover its real values on Hermitian elements.
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