In many parts of analysis an important role is played by multilinear maps. Recall that if E, F, and Z are vector spaces, then a map γ: E × F → Z is bilinear provided that it is linear in each variable, i.e., γ(e1 + e2, f1) = γ(e1, f1) + γ(e2, f1), γ(e1, f1 + f2) = γ(e1, f1) + γ(e1, f2), and γ(λe1, f1) = γ(e1, λf1) = λγ(e1, f1) for any e1 and e2 in E, for any f1 and f2 in F, and for any λ in. If one forms the algebraic tensor product E ⊗ F of E and F, then there is a one-to-one correspondence between linear maps Г: E ⊗ F → Z and bilinear maps γ: E × F → Z given by setting Г(e ⊗ f) = γ(e, f).

Consequently, if one endows E ⊗ F with a matrix norm, then the completely bounded linear maps from E ⊗ F to another matrix-normed space Z correspond to a family of bilinear maps from E × F to Z that one would like to regard as the “completely bounded” bilinear maps. In this fashion, one often arrives at an important family of bilinear maps to study.