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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
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exp(S) exp(T) = exp(S+T) for commuting operators
The standard way to prove the exponential law for two bounded commuting operators $S, T$
$$
\exp(S)\exp(T) = \exp(S+T)
$$
is to pass by the binomial formula and the power series of $\exp(.)$. I wonder …