Introduction

The result known as Grothendieck's inequality is coming to be recognized as one of the really major theorems of Banach space theory. It first appeared in Grothendieck (1956) under the title “the Fundamental theorem of the metric theory of tensor products” (as we have seen, a number of the other results considered in this book can be traced to the same memoir). In fact, the theorem admits a remarkable number of equivalent formulations, expressed variously in terms of summing norms, bilinear forms and tensor products. One version says that J?1 has a property rather stronger than being 2-dominated. Some of these formulations were given by Grothendieck himself, others by later writers. A particularly elementary version was given by Lindenstrauss & Pelczynski (1968); this served to make the theorem much better known and understood by mathematicians generally.

The theorem has many applications, both within Banach space theory and in other areas, notably harmonic analysis (we cannot attempt to do justice to these in this book). Also, there is by now a repertoire of alternative proofs that must have few parallels in Mathematics. Despite this, the exact determination of the constant appearing in the inequality remains an unsolved problem. There are actually two separate problems, for the real and complex cases respectively.

In this section, we start with the Lindenstrauss-Pelczynski formulation, and give a version of the proof, due to Krivine (1979), that yields the best current estimate of the constant in the real case.