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.

The summing and nuclear norms of linear operators merit recognition as very basic concepts in Banach space theory, even at quite an elementary level. They have powerful applications to a variety of Banach space questions, and they generate a theory that is interesting and elegant in its own right. It is hoped that the pages that follow will go some way towards justifying these assertions. The only prerequisite is a beginner's course on normed linear spaces. As well as the confirmed Banach space specialist, our topic has something to offer to analysts whose main interest is, for example, approximation theory or operator theory.

The origins of the subject can be traced to Khinchin's inequality (published in 1923) and to Orlicz's deduction (1933) that for every unconditionally convergent seriesΣxn in Lp (where 1 ≤ p ≤ 2), σ ‖xn‖2 is convergent. In 1947, Macphail showed that in such a series may have σ ‖xn‖ divergent. Dvoretzky and Rogers then proved that the same applies in every infinite-dimensional Banach space. From this, it was a short step to define an “absolutely summing operator” to be one for which σ∥Txn∥ is convergent for every unconditionally convergent series σxn. Further, Macphail's work showed how this property is equivalent to a certain numerical quantity being finite: this is the “1-summing norm” π1(T). The idea generalizes easily to give norms πp for each finite p ≥ 1.