This paper asks: given a vector bundle ξ and a line bundle λ over the same base space, are λ[otimes ]ξ and ξ equivalent? We concentrate on real bundles ξ. Although the question is sensible in its own right, we explain in Section 2 our immediate motivation for studying it. In Section 3 we make some general comments about the question, the most significant being that under certain restrictions the answer depends on the stable class of ξ rather than on ξ itself (Proposition 3·4).

The rest of the paper tackles an interesting special case. To state the main result, let P(ℝn+1) denote n-dimensional real projective space, H the Hopf line bundle over it, and an+1 the order of the reduced Grothendieck group [wavy overbar]KO0(P(ℝn+1)).