I'm comfortable with the idea that a vector space $V$ and its double-dual $V^{**}$ are naturally isomorphic, independent of bases: $$ \Phi:V \to V^{**}, \quad v \mapsto \hat{v} $$ $$ \quad \big( \Phi(v) \big)(f) = \hat{v}\,(f) = f(v) $$ $$ \forall f\in V^*, \forall v\in V $$
It is unclear to me, however, why $V$ and $V^*$ are not naturally isomorphic. It's clear that they can be made "unnaturally" isomorphic (if you will) by choosing a particular set of bases and cooking up the trivial isomorphism between these two sets. What is less clear is why there cannot be a natural isomorphism between them.
Is this some kind of category theoretic result? It seems proving that no natural isomorphism could ever exist is a much taller order than showing one particular isomorphism does exist. Any thoughts?
