Theorem 9.3.6 (Stable complements, Hatcher (2003), Proposition 1.4). Let \(X\) be a compact Hausdorff space. For every complex vector bundle \(E \to X\) there exists another vector bundle \(E' \to X\) such that the direct sum \(E \oplus E'\) is isomorphic to a trivial vector bundle \(\ul {\C }^N := X \times \C ^{N}\) for some \(N \in \N \).
Proof sketch. Choose finitely many local trivializations and a subordinate partition of unity. Multiplying the local coordinate functions by the partition-of-unity functions embeds \(E\) into a trivial bundle \(X \times \C ^N\). After choosing a Hermitian inner product on this trivial bundle, the orthogonal complements of the fibers of \(E\) assemble to a vector bundle \(E'\) with \(E \oplus E' \cong \ul {\C }^N\). The details are the usual compactness and partition-of-unity argument; see Hatcher for the full proof. □
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