I feel like this has to have been asked before, but my searches turned up nothing.
I was tutoring a student today and they asked me what is a good intuition for an adjoint. I still am not sure I know the answer to that question, but I figured I would think it through. I know there is a strong relationship between inner products and adjoints (by definition) and inner products and dual spaces (by Riesz).
However, it occurred to me as I was trying to sort this out in my mind: I have no idea what is a good definition of the inner product on a dual space. That is, if $V$ is a Hilbert space, how do we use the inner product of $V$ to create an inner product on $V^*$? Surely one must exist, but I couldn't think up an obvious example, except by passing through Riesz. Is there something more elementary?
[Side Note: I'm not sure how much I like adjoint as a tag, but okay]