@leslietownes Though in a lot of cases, those won't really be inner products, but the dual pairing (which devolves to an inner product in spaces which are self-dual. (Not that this really matters).
@leslietownes Actually sorry for perseverating on this Leslie, but I've been searching around and not able to find a wikipage or something like that for this. What term should I be using to learn more about this equivalence between the two possible inner products on this periodic function space?