I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 72 is this theorem:
And the proof gose like this:
And so on. My question is at the second to last proposition. namely the fact that we cand get a local basis of section for $E.$ Is this something more general like for vector bundles? I'm new to the subject.
Given a bundle $\pi: E \rightarrow M$, by definition you have trivializations, that is maps $t_U:\pi^{-1}(U) \rightarrow U \times V$ where $V$ is the vector space on which the bundle is modeled. Also for all $x\in U$ $t_U(\pi^{-1}(x))\simeq V$. So take a base $\lbrace e_i \rbrace$ of $V$, and consider sections $\sigma_i:U\rightarrow\pi^{-1}(U):x\mapsto (x,t_U^{-1}(e_i))$ then those are locally a base.