 Classification of complex vector bundles by a Grassmannian

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.