Let $\pi: E \to M$ and $\pi': E' \to M$ be two real vector bundles and denote the set of all linear maps between the fibres $E_p$ and $E'_p$ by $\text{Hom}_{\mathbb R}(E_p,E'_p)$. I want to show that the disjoint union$$\text{Hom}_{\mathbb R}(E,E') := \bigsqcup_{p \in M} \text{Hom}_{\mathbb R}(E_p,E'_p)$$ is a real vector bundle itself. Let's denote the elements of $\text{Hom}_{\mathbb R}(E,E')$ as pairs $(p,L)$ where $p \in M$ and $L \in \text{Hom}_{\mathbb R}(E_p,E'_p)$.
An obvious candidate is the surjection $\sigma: \text{Hom}_{\mathbb R}(E,E') \to M,\ \ (p,L) \mapsto p$. So we are left to show that $\sigma$ is continuous and that there are local trivialisations.
My problem is that I don't know what the topology on $\text{Hom}_{\mathbb R}(E,E')$ is and that I don't know any book which covers this example.
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Philipp
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Your map $\sigma$ is correct: you basically define the topology on the Hom bundle so that this projection is continuous: if $U$ trivializes $E$ and $E'$ so that $E_U,E'_U \cong V,V'$ respectively then $\sigma^{-1}(U) \cong U\times \textrm{Hom}(V,V')$. Then one has to check these sets glue to define a coherent global topology. Details can be found in http://virtualmath1.stanford.edu/~conrad/diffgeomPage/handouts/bundleops.pdf – wskrsk Jun 09 '18 at 20:35
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Nice, exactly what I was looking for. Thanks. – Philipp Jun 10 '18 at 07:23