Lee, Introduction to smooth manifolds, defines a vector bundle $\pi :E \rightarrow M$ via some local trivialization maps $\pi^{-1}(U) \rightarrow U \times R^k$, where $U \subset M$.
There is something about this that is a bit hindering my intuition. Do I have to consider $R^k$ here as some coordinates of a vector space, or as a fixed vector space itself? Why did not Lee define just a fixed a vector space $V$ and define trivializations as functions $\pi^{-1}(U) \rightarrow U \times V$?.
A vector space $V$ is still a manifold and in some sense this definition is less confusing for me because $R^k$ can be either considered as a fixed vector space, or some coordinates after fixing a base in $V$... $U \times R^k$ looks to me a bit weird since $U$ is just a subset of a manifold, whereas $R^k$ looks like some coordinates in a chart.
Please be patient and consider that I am still trying to build a basic understanding/intuition..
When I studied vector bundles, the definition my professore gave was with a generic vector space $V$.
I think Lee decided to go with $\mathbb{R}^k$ in order to emphasize the fact that it is a real vector space of dimension $k$.