I'm a bit confused about the following part in Sir Michael Atiyah's "K-Theory."
Let $E = X \times V$ and $F = X\times W$, and let $\phi: E\to F$ be a vector bundle homomorphism. Why is the induced map $\Phi: X \to Hom(V,W)$ continuous? $V$ and $W$ are finite dimensional vector spaces (over $\mathbb{C}$).
To be a bit more precise, for $(x,v) \in E$, $\phi(x,v) = (x,\Phi(x)v)$.