Classification of complex vector bundles by a Grassmannian (${\mathrm{Vect}}_{\mathbb{C}}^k(X)\cong [X,{\mathrm{Gr}}_k({\mathbb{C}}^{\infty })]$)

A partition of unity and local orthonormal frames embed a bundle over compact Hausdorff X in a finite trivial bundle. Its fibre images give a classifying map pulling back the complex tautological bundle on a Grassmannian. Isomorphic bundles give homotopic maps by interpolating two embeddings in orthogonal coordinate blocks. Homotopic maps give isomorphic bundles because uniformly close orthogonal projections identify their image bundles along a finite subdivision of the homotopy interval.

The contractible universal frame bundle gives ${\pi }_4(B{U}_2)\cong {\pi }_3(U_2)$ and simple connectedness of $B{U}_2$. The determinant fibration $S{U}_2\to U_2\to S^1$ identifies ${\pi }_3(U_2)$ with ${\pi }_3(S{U}_2)$. Since $S{U}_2\cong S^3$, this is $\mathbb{Z}$. Hence the classification of complex vector bundles by a Grassmannian gives the displayed bijection of isomorphism classes; fixed-rank classes do not acquire a group operation from Whitney sum.

Let P,Q be continuous orthogonal projection matrices of equal rank on a common trivial complex vector bundle, with the displayed bound at every base point. Restricting Q to the image of P is injective: if Qv=0, then $v=(P-Q)v$ contradicts the norm bound unless v=0. Equal dimensions give a continuously invertible vector bundle isomorphism. Compactness applies this observation uniformly to a homotopy of image planes.

For finitely many orthonormal local trivializations $t_i$ and a subordinate partition of unity ${\rho }_i$, the displayed map embeds a rank-k complex vector bundle isometrically in $X\times {\mathbb{C}}^{kr}$. Outside the support of each ${\rho }_i$ its component is zero. Its continuous image planes give a finite-dimensional Grassmannian map and identify the original bundle with a tautological pullback.