Subbundle of vector bundle over paracompact admits complement

This article describes a result applicable for real vector bundles over a paracompact Hausdorff space. In particular, the result is applicable for real vector bundles over a manifold, CW-space, or metrizable space

Statement

Any vector subbundle of a real vector bundle over a paracompact Hausdorff space, admits a direct complement. In other words, there is another subbundle such that at every point, the sum of the fibers for the two subbundles equals the fiber for the whole bundle.

Proof

The proof has the following steps:

• Use the fact that any vector bundle over a paracompact base admits an inner product
• Consider, fiber-wise, the orthogonal complement to a given subbundle of the vector bundle
• Prove that this fiber-wise orthogonal complement is actually a subbundle