How to view complex vector bundle as a real vector bundle

Question

I'm trying to understand how to view complex vector bundles as real vector bundles.

I am confused about what is going on in the wikipedia page for complex vector bundles. Here is a screenshot:

In particular I am confused about the sentence:

If $E$ is a complex vector bundle, then the complex structure $J$ can be defined by setting $J_x$ to be the scalar multiplication by $i$.

Here $E$ is a complex vector bundle, but $J$ is defined on real vector bundles. What should be the domain and target of $J_x$ in this case?

Answer 1

Any complex vector space is also a real vector space, by just restricting the scalar multiplication operation to just real scalars. In the same way, any complex vector bundle is also a real vector bundle, by considering each fiber as a real vector space. (To see that this bundle of real vector spaces is locally trivial, note that any local trivialization of the complex vector bundle also gives a local trivialization as a real vector bundle by identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ in the obvious way.)