Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$.
The definition as stated require us to start with an inner product $(.,.)$ in $V$ and check if the operator $A$ satisfies the equality.
My question is:
If we start with an operator $B$ on a vector space $W$ what are the necessary and sufficient conditions such that we can define an inner product such that $B$ is self adjoint with that inner product?