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Use the customary compact Hausdorff base convention for the classification of complex vector bundles by a Grassmannian; equivalently the argument uses finite subordinate partitions of unity. Give a rank-k complex vector bundle

E \to X

a continuous fiber metric given by Hermitian inner products. Choose a finite trivializing cover with local orthonormal coordinates

t_{i} : E ∣_{U_{i}} \to U_{i} \times C^{k}

, and a subordinate partition of unity

ρ_{i}

with support contained in

U_{i}

. Define

J_{x} (v) = (ρ_{i} (x) t_{i, x} (v))_{i} \in C^{k r},

(2)

extending each component by zero off its support. This is continuous and satisfies

∥ J_{x} (v) ∥^{2} = \sum_{i} ρ_{i} (x) ∥ v ∥^{2} = ∥ v ∥^{2}

, so it is a fibrewise embedding. The image planes vary continuously: in local frames, their orthogonal projection matrix is

J (J^{*} J)^{- 1} J^{*}

. Thus they define a map

c_{E} : X \to Gr_{k} (C^{k r})

, and

v \mapsto (x, J_{x} v)

identifies

E

with

c_{E}^{*} γ_{k}^{\infty}

. This proves that every bundle is a tautological pullback vector bundle, by the finite-dimensional embedding of a complex vector bundle.

We next prove the two independence assertions that turn this construction into a bijection. First, homotopic maps have isomorphic pullback vector bundles. Pulling back along a homotopy gives a bundle over compact Hausdorff

X \times I

; embed that bundle in a finite trivial vector bundle by the construction just given. Its image has a continuous family of rank-k orthogonal projection matrices

P (x, t)

. Compactness gives a subdivision

0 = t_{0} < \dots < t_{s} = 1

with

∥ P (x, t_{j + 1}) - P (x, t_{j}) ∥ < 1 for every x .

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Restriction of

P (x, t_{j + 1})

to the image at

t_{j}

is injective: if it killed

v

, then

v = (P (x, t_{j}) - P (x, t_{j + 1})) v

, impossible by the strict norm bound unless

v = 0

. Equal ranks make it an isomorphism, with continuous local inverses. Composing these bundle maps identifies the endpoint vector bundles. The fact that close orthogonal projections identify their image bundles therefore makes the pullback assignment well-defined on homotopy classes.

Conversely, suppose two classifying maps give isomorphic vector bundles. The infinite Grassmannian has its standard weak CW topology; the compact-subset property of CW complexes puts the compact images of the two maps in finite stages. Their tautological pullback vector bundles therefore give two finite-dimensional embeddings

J_{0} : E \to C^{N_{0}}

and

J_{1} : E \to C^{N_{1}}

of the same bundle, after using the vector bundle isomorphism. In orthogonal coordinate blocks interpolate by

J_{t} (v) = (cos \frac{π t}{2} J_{0} (v), sin \frac{π t}{2} J_{1} (v)) .

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At least one coefficient is nonzero, so every

J_{t}

is injective and its image planes give a continuous Grassmannian homotopy. One endpoint uses the first block and the other the second block. Moving the latter coordinate copy to the standard one is a fixed unitary change of coordinates, connected to the identity by a path in the unitary group. Such a path exists by diagonalizing the unitary matrix and continuously varying its eigenvalue phases. Hence the original two maps are homotopic in the infinite Grassmannian. This also proves independence of all embedding and metric choices. Together the two directions establish

Vect_{C}^{k} (X) ⟷ [X, Gr_{k} (C^{\infty})], [f] ⟼ [f^{*} γ_{k}^{\infty}] .

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For arbitrary non-Hausdorff compact spaces without the required numerability, that usual bundle-classification statement needs additional hypotheses; the construction above specifies precisely the conventional setting being used.

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