I'm trying to find an explicit expression for the adjoint of a particular linear operator. Are there any very general techniques for doing so when we know very little about the setting we're working in? All I have is that

is a linear operator and that it does have an adjoint. I know nothing about

(inner product, dimension (possibly infinite), etc.).

The book doesn't give any examples like this, and the only one I've found online uses

, so we can work with the inner product formulas explicitly to obtain the adjoint.