This might be a simple question,but I wonder where I did wrong.I have seen this: Why aren't complex vector bundles isomorphic to their duals? .
Let $E$ be a complex vector bundle with an hermitian structure $h$,we know that as real bundles,we have $E\cong E^*$.However,as complex vector bundles,it is incorrect in general.
But we always have $\overline E\cong E^*$.Here is the proof:since $h(v,w)$ is linear in $v$ and antilinear in $w$,the map $w\longmapsto h(\cdot,w)$ defines an isomorphism from $\overline E$ to $E^*$.
Then,if I define $v\longmapsto h(v,\cdot)$,isn't it an isomorphismfrom $E$ to $E^*$?
Thanks in advance.