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In general it is not true that a vector bundle $E$ is isomorphic to its dual bundle $E^*$. But it is true when the vector bundle is the tangnet space of a manifold (at least I think it's true). How does one prove this?

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up vote 10 down vote accepted

An isomorphism between a vector bundle and its dual, fibre-by-fibre is just an isomorphism between a fibre and its dual. An isomorphism between a vector space and its dual is given by a non-degenerate bilinear function -- so for example, an inner product suffices. So if you had an inner product on your vector bundle, you would have an isomorphism between $E$ and $E^*$ simply by the operation $v \longmapsto \langle v, \cdot \rangle$.

Generally speaking, vector bundles have inner products. For example, if the base space is paracompact and the fibres are finite-dimensional.

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Pick a metric on $M$ and use it to identify each tangent vector space to its dual. This gives a smooth isomorphism $TM\cong T^*M$.

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