I am reading Vector Bundles and the Kunneth Formula and in the proof of lemma 2, Atiyah states $\widetilde K ^0(X) = [X,BU]$ and $K^1(X)=[X,U]$ without justification. Can someone give me a reference for this result?
Is this related to the classification theorem for unitary bundles: $K(X) \cong [X, \mathbb Z \times BU]$ and $\widetilde{K}(X) \cong [X, \mathbb Z \times BU]_0$, where $[X,Y]_0$ are homotopy classes of base point preserving maps. If so, how?
It should be something like this. By definition $K^1(X)=\tilde K(\Sigma X_+)=[S(X_+),\mathbb{Z}\times BU]_0$, where $\Sigma$ means the suspension and $X_+$ is a space with a new basepoint added. Using the suspension loop space adjunction and the fact that the based loop space only cares about the component of the basepoint this set equals $[X_+,\Omega BU]_0$. Now the basepoint is mapped to a basepoint of $\Omega_BU$ so we can forget about it since we only care about $X$. Hence $K^1(X)=[X,\Omega BU]$.
