Naturality of Riesz' Representation What does it mean precisely in the context of category theory when somebody says that Riesz' representation is **canonic** resp. every Hilbert space is **naturally** antiisomorphic to its dual?

This is a very interesting issue. In fact, this isomorphism is _not_ as "natural" as one might have thought. As an exercise, one should see that, given a map of Hilbert spaces $V\to W$ it is rarely the case that the square of maps involving $W^*\to V^*$ and the "Riesz-Fisher" dualities ... commutes. This is fairly crazy, yes, given the standard curriculum.

A similar deceptive fake-issue arises while thinking about Sobolev spaces.