The revolution in algebraic geometry and homological algebra wrought by Alexandre Grothendieck is arguably the defining feature of at least one major trend in late twentieth century mathematics. Perhaps this trend can be usefully described as a shift of focus away from traditional mathematical objects to the relations and mappings that hold between classes of such mathematical objects — or more exotic and novel ones. Certainly this appraisal of Grothendieck’s philosophy and, concomitantly, his enthusiastic embrace of category theory, is not new; indeed, if I may be forgiven a flippancy, Grothendieck’s very approach to algebraic geometry suggests that for him it was at least as much about the arrows (i.e. morphisms) between the objects as about the objects themselves. Additionally, Grothendieck himself likened his methods to a steadily rising sea eventually submerging the island that is the problem at hand, and, accordingly, the problem is solved by its “disappearance” in the ambient theory. (I think this metaphor is available somewhere in Grothendieck’s controversial Récoltes et Semailles, his relatively recent memoirs containing sundry reflections on his life in mathematics, and much else besides). Thus we have Grothendieck’s restructuring of homological algebra in his famous Tôhoku paper as the means whereby to immerse algebraic geometry in a sea of new techniques, with many spectacular successes, providing an exemplar of a unique and idiosyncratic working-style all Grothendieck’s own: he approached his work much like an architect, first sketching a vast project, drafting parts of it in greater detail than otherwise, giving it all shape early on, and all of it meant as a prelude to huge and diversified later efforts aimed at crafting a grand and far-reaching theory. Certainly Grothendieck’s work in algebraic geometry fits this description, particularly as regards the S(éminare de) G(éométrie) A(lgébrique du Bois-Marie) and the E(léments de) G(éométrie) A(lgébrique) series. It is frequently speculated that the sheer size of these undertakings contributed critically to Grothendieck’s departure from the mathematical scene: after many years of non-stop labor he realized that he’d only just begun; the claim is made that this is why EGA runs to only a couple of finished volumes and is left forever incomplete.
The point is that Grothendieck’s scholarly style was designed with a special orbit in mind: it was centered on grand conceptions, it was possessed early on of considerable structure, and it was intended to lead to major future activity. Additionally, it is a marvelous circumstance that this was so already at the start of Grothendieck’s career, some years before he was bewitched by algebraic geometry. Grothendieck’s first work occurred in analysis, specifically functional analysis.
The book under review, The Metric Theory of Tensor Products: Grothendieck’s Résumé Revisited, is concerned with a famous blue-print (and then some!), the Résumé, that Grothendieck composed in São Paolo in the early and middle 1950s. The authors, Diestel, Fourie, and Swart, introduce their “revisit” to Grothendieck’s early work as follows: “His inborn compass led him to isolate notions that were to play a central role in the study and the development of Banach space theory to this very day. He was the first to formulate isomorphic invariants of special Banach spaces by comparing these spaces with other Banach spaces via the bounded linear operators between them… He recognized the importance of the nature and location of the finite dimensional subspaces of a space and utilized such — ‘local’ theory was born. He was the first analyst to seriously chase diagrams in the hopes of catching essential isomorphic characteristics of Banach spaces, and catch them he most certainly did. Nowhere are there innovations more in evidence than in his infamous Résumé.”
The book starts off with a long chapter on the “[b]asics on tensor norms,” goes on to discuss C(K)-spaces and L1-spaces, “⊗-norms related to Hilbert space,” and then reaches the culmination of these themes in the fourth chapter on Grothendieck’s fundamental inequality (“Lindenstrauss-Pelczynski style” (p. 192)). There are four relatively ramified appendices, dealing, for instance, with Banach lattices (including a section titled “The facts ma’am, just the facts” — humor is always a plus when it comes to functional analysis. Along these lines I also want to direct the reader to p. 100 of the book, where we find the fragment: “Y**** (yikes!)”), and a section titled “Comparison of the projective and injective tensor products” which certainly looks like a harbinger of things to come, what with the future prominence of derived categories as launched and developed by Grothendieck and his pupils in algebraic geometry. This surely is a vindication of Grothendieck’s methods — not that they needed it, of course.
Thus, The Metric Theory of Tensor Products scores on several counts, not just as a serious scholarly contribution to functional analysis, but as a tribute to Grothendieck’s incomparable gifts in the area of innovation and originality.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.