I am reading K-theory by M.Atiyah and am having difficulty in understanding the proof of Bott periodicity. On page 72, he mentions that the homomorphism $\alpha : K(S^2 \times X) \to K(X)$ is a $K(X)$-module homomorphism. I do not understand : how is $K(S^2 \times X)$, a $K(X)$ module in the first place ? Is there some standard way of doing it ?
I can think of the following : There is a projection map $\pi_2 : S^2 \times X \to X$. This shall induce a ring homomorphism $\pi_2^*: K(X) \to K(S^2 \times X)$. Thus one can think of $K(S^2 \times X)$ as a $K(X)$-module. Is this the module structure that the book refers to ?