This question is from a passage in the beginning of Atiyah's K-Theory notes.
Suppose that $X$ is a topological space and $E=X \times V$ and $F= X \times W$ are two product bundles, where $V$ and $W$ are topological, finite dimensional vector spaces. Clearly, a bundle homomorphism $\phi: E \to F$ corresponds to a map $\Phi: X \to \mathrm{Hom}(V,W)$ by the formula $\phi(x, v) = (x, \Phi(x)v)$.
Atiyah claims that equipping $\mathrm{Hom}(V,W)$ with "the usual topology" makes $\Phi$ continuous. I'm trying to figure out what topology he means. My naive guess was the compact open topology, but I couldn't figure out how to show that $\Phi$ was continuous in this case. I noticed that for every $v \in V$, the map $x \mapsto \Phi(x)v$ is continuous. Could Atiyah be talking about a weak topology of some kind?
If it helps, later on in the passage he claims that $\mathrm{Iso}(V,W)$, the set of linear isomorphisms between $V$ and $W$ is an open subset of $\mathrm{Hom}(V,W)$ with this topology.