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Is there an easy way to see the equivalence of the two statements of Bott periodicity via K-Theory and the original Morse Theory proof?

So, $$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and $$K(X)\otimes K(S^2) \cong K(X\times S^2)$$

I know there is the Bott paper that explicitly comments on this, but it is written in French. Any help will be greatly appreciated!

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  • $\begingroup$ This should help at least a little: ncatlab.org/nlab/show/… $\endgroup$
    – David Roberts ♦
    Commented Aug 4, 2017 at 1:35
  • $\begingroup$ Considering the reference to Bott's paper, it's probable the OP was asking a more subtle question: why does the Bott map (something defined in terms of grassmanians and paths) agree with multiplication by the Bott class? In other words, there's a little diagram of isomorphisms and it's not at all obvious it commutes... I think Bott's paper checks this. $\endgroup$
    – Dylan Wilson
    Commented Aug 4, 2017 at 13:46

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