Hi, please give me some leeway for my laziness here: We have that , in the finite-dimensional case for vector spaces, V~V** in a natural, i.e., basis-independent way ; one way of proving this is by showing that the identity functor is naturally-isomorphic to the double-dual functor. An easy way of showing that this is not the case for V~V* is that the two functors ( identity and 1st-dual ) go in opposite directions. Questions: 1) Is there a canonical bilinear form B:V-->V** here that gives us the isomorphism V~V**? How do we show there is no such form from V to V*? 2) How/why does the functor argument showing V~V** fail when V is infinite-dimensional? Thanks for answers, hints, refs.