Prove AB is hermitian if A is hermitian and B is hermitian If $A$ and $B$ are two hermitian transformations, prove that $AB$ is hermitian if $AB = BA$, knowing that a hermitian transformation is one such that $(T(f), g) = (f, T(g))$ and basic axioms for inner products: $(x,y) = (y,x)$, $(x,x) > 0$, $(cx, y) = c(x,y)$, and $(x,y+z) = (x,y) + (x,z)$.

$$\langle ABf, g \rangle = \langle Bf, Ag \rangle = \langle f, BAg \rangle = \langle f, ABg \rangle.$$
Here


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*I "=" $\longleftarrow$ $A$ is Hermitian,

*II "=" $\longleftarrow$ $B$ is Hermitian,

*III "=" $\longleftarrow$ $AB = BA$.

