What kind of tensor is the gradient of a vector Field?

In summary, The gradient of a vector field is a (1,1) tensor and is represented by a smooth section of the tensor product bundle of the tangent bundle and the dual bundle. This gradient can be thought of as a function of the direction vector and is linear in that it factors over linear combinations of vector fields. This formalism can be extended to more general vector bundles by considering connections, which are linear mappings that satisfy the Leibniz rule.
  • #36
S.G. Janssens said:
As an example, take a Banach space that is not reflexive but still linearly isomorphic to its second continuous dual space.
How is that in contradiction to what I said!
 
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  • #37
martinbn said:
How is that in contradiction to what I said!
Is the above a question?

If yes, then it can be answered by looking up reflexivity and, for example, James' theorem.
 

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