Let $A, B:V \to V$ be two linear operators in a vector space with inner product. If $A$ and $B$ commutes then so does $A^\ast$ and $B^\ast$

Question

$A^\ast$ and $B^\ast$ are adjoint operators

This is a proof that is for an undergraduate linear algebra class. This is what I tried but I don't know if it is enough or rigorous enough.

$\langle Av, w\rangle = \langle w, A^\ast w\rangle$

$\langle ABv, w\rangle =\langle w, A^\ast B^\ast w\rangle$

$AB = BA$

then

$\langle BAv, w\rangle = \langle w, B^\ast A^\ast w\rangle$.

Answer 1

Hint: What is the adjoint of $ST$, for operators $S,T$?