A bilinear map ø Ra x Rb → Rc is non-singular if ø (x, y) = 0 implies x = 0 or y = 0. For background information on such maps see (4, 5, 6, 14). If we apply the ‘Hopf construction’ to ø, we get a map
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100054086/resource/name/S0305004100054086_eqnU1.gif?pub-status=live)
defined by
2ø(x, y)) for all x ∈ Ra, y ∈ Rb satisfying ∥x∥2 + ∥y∥2 = 1. Homotopically, Jø coincides with the map obtained by first restricting and normalizing ø to
, and then applying the standard Hopf construction ((13), p. 112). In any case, one gets an element [Jø] in
, which in turn determines a stable homotopy class of spheres {Jø} in the d-stem
, where d = a + b − c −1. An element in
which equals {Jø} for some non-singular bilinear map ø will be called bilinearly representable. The first purpose of this paper is to prove