For \(k = 1\) the classifying space
\begin{equation*}
BU(1) \;=\; \mathrm{Gr}_1(\mathbb{C}^\infty) \;=\; \mathbb{C}P^\infty
\end{equation*}
is the infinite complex projective space. For a CW complex \(X\text{,}\) Theorem 17.2 therefore gives
\begin{equation*}
\mathrm{Vect}_1^{\mathbb{C}}(X) \;\cong\; [X,\, \mathbb{C}P^\infty].
\end{equation*}
Since \(\mathbb{C}P^\infty = K(\mathbb{Z}, 2)\) is an Eilenberg–MacLane space, Theorem 14.9 further identifies
\begin{equation*}
\mathrm{Vect}_1^{\mathbb{C}}(X) \;\cong\; [X,\, K(\mathbb{Z}, 2)] \;\cong\; H^2(X; \mathbb{Z}).
\end{equation*}
The class corresponding to a line bundle \(L\) is its first Chern class \(c_1(L) \in H^2(X; \mathbb{Z})\text{.}\) So line bundles, which are geometric objects, are completely determined by a cohomology class. One can show that \(c_1(L \otimes L') = c_1(L) + c_1(L')\text{,}\) so this bijection is a group isomorphism when \(\mathrm{Vect}_1^{\mathbb{C}}(X)\) is given the tensor product.
