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

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Discussion Overview

The discussion revolves around the isomorphism between the dual vector bundle E* and the space Hom(E, MXR). Participants explore definitions, properties, and potential isomorphisms related to dual vector bundles, with a focus on the mathematical structures involved.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants reference the definition of E* as Hom(E, MXR) and seek clarification on the nature of the isomorphism between these two structures.
  • One participant questions the definition of E* and emphasizes the importance of understanding it before discussing isomorphisms.
  • Another participant mentions a natural isomorphism involving Hom(E, E') and suggests that this could lead to an isomorphism between E* and Hom(E, MXR), although the reasoning is not fully established.
  • There is a claim that the dual of a vector bundle is obtained by taking fiber-wise duals, with a specific reference to the fibers of E* being the duals of the fibers of E.
  • One participant expresses confusion about the discussion, suggesting that the arguments presented may be tautological.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the definitions and implications of the isomorphism between E* and Hom(E, MXR. There are competing views on how to define E* and the nature of the isomorphisms involved.

Contextual Notes

There are limitations regarding the clarity of definitions and the assumptions underlying the proposed isomorphisms. The discussion does not resolve these ambiguities.

robforsub
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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)?
 
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So how do you define E*?
 
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)?
 
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).
 
I'm failing to see how it's not tautological.
 

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