= Classification of complex vector bundles by a Grassmannian
{title2=$\operatorname{Vect}_{\mathbb C}^k(X)\cong[X,\operatorname{Gr}_k(\mathbb C^\infty)]$}
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.
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