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.