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 $(x,y+z) = (x,y) + (x,z)$.

We prove that $(AB)^*=AB$ by these equalities
$$\langle x,(AB)^*y\rangle=\langle ABx,y\rangle=\langle Bx,A^*y\rangle=\langle Bx,Ay\rangle=\langle x,B^*Ay\rangle=\langle x,BAy\rangle=\langle x,ABy\rangle$$
