dual vector bundle E* is isomorphic to Hom(E, MXR)

robforsub

As E* is defined in some book as Hom(E, MXR). What could be the isomorphism between dual vector bundle E* and Hom(E, MXR)?

quasar987

So how do you define E*?

robforsub

That's the part that is confusing to me. So I have checked on wikipedia, and it defines E*=Hom(E,MXR). However, there is a natural isomorphism on bundle that is Hom(E,E')=E*(direct sum)E, therefore I am wondering if I can use this isomorphism to get the result that E* is isomorphic to E*(direct sum) MXR and thus isomorphic to Hom(E, MXR)?

Landau

Still, how does your question even make sense if you have no definition of E*? For me, the dual of a vector bundle E is obtained from E by taking fiber-wise duals, i.e. the fibers of E* are the vector space duals of the fibers of E. Of course, the dual of a k-vector space Ep is Ep*=Hom_k(E,k).

zhentil

I'm failing to see how it's not tautological.