##
**Introduction to tensor products of Banach spaces.**
*(English)*
Zbl 1090.46001

Springer Monographs in Mathematics. London: Springer (ISBN 1-85233-437-1/hbk). xiv, 225 p. (2002).

The study of norms on the tensor product of Banach spaces goes back to the work of F. J. Murray and J. von Neumann in the late 1930’s, and in particular to the work of R. Schatten which appeared about ten years later. Apparently unaware of the latter, A. Grothendieck started his own investigations on tensor norms and applications in his fundamental ‘Résumé de la théorie métrique des produits tensoriels topologiques’ [Bol. Soc. Mat. São Paulo 8, 1–79 (1956; Zbl 0074.32303)]. Together with ‘Produits tensoriels topologiques et espaces nucléaires’ [Mem. Am. Math. Soc. 16 (1955; Zbl 0123.30301)], the ‘Résumé’ has deeply influenced developments in functional analysis, and in particular in Banach space theory.

Grothendieck formulated his results using the language of tensor norms. Proper understanding of the involved concepts, however, requires much more than just some acquaintance. As a result, Grothendieck’s work, in particular his ‘théorème fondamental de la théorie métrique de produits tensoriels’, remained largely unnoticed for many years. This changed only in 1968 when J. Lindenstrauss and A. Pełczyński published their seminal paper [Stud. Math. 29, 275–326 (1968; Zbl 0183.40501)] in which they provided a matricial formulation of the ‘théorème fondamental’, nowadays known as Grothendieck’s inequality, and linked and extended many of Grothendieck’s results by using the theory of absolutely summing operators, as developed just one year earlier by A. Pietsch [Stud. Math. 28, 333–353 (1967; Zbl 0156.37903)]. In the sequel, the local theory of Banach spaces, for example, developed largely along these lines. However, up to few exceptions, the viewpoint of tensor norms was given up in favour of the easier accessible one of ideals of operators and their natural norms [cf. A. Pietsch’s monograph “Operator Ideals” (North–Holland Mathematical Library 20) (1980; Zbl 0434.47030)]. Several attempts have been undertaken to gain additional insight into topics like local Banach space theory by re-incorporating tensor norms into the developments. Most remarkably in this regard is the monograph by A. Defant and K. Floret [“Tensor Norms and Operator Ideals” (North–Holland Mathematics Studies 176) (1993; Zbl 0774.46018)].

Whereas the latter is a fairly advanced book, the one under review is intended to serve as an introduction to the theory of tensor products of Banach spaces. As such, it is a most welcome addition to the existing literature and appears to be well-suited as a guide and as a textbook in lectures, seminars, etc., for students having the standard facts from functional analysis and measure theory at their disposal.

The book starts by introducing, in the first chapter, tensor products of two vector spaces by means of ‘point evaluations’ on the corresponding space of bilinear forms; the basic algebraic facts are derived in a straightforward manner. Projective and injective tensor norms form the subject matter of the next two chapters, with emphasis on relations to Bochner and Pettis integrals, and to nuclear and integral operators, respectively. The fourth chapter deals with the approximation property and related matters. Among the topics which are discussed are reflexivity of spaces of operators, of tensor products, bases in tensor products of spaces with a basis, etc.

Chapter 5 is devoted to the Radon–Nikodym property for Banach spaces, to spaces of vector measures in comparison to projective and injective tensor products of spaces of scalar measures with the Banach space under discussion, and to the principle of local reflexivity. Chapter 6 starts with a brief introduction to general tensor norms. The main examples treated here are the Chevet–Saphar norms; it is explained how they lead to the ideals of \(p\)-summing operators. The standard results for such operators, including Grothendieck’s inequality, are derived. Chapter 7 compares Schatten’s ‘dual tensor norm’ with the corresponding one introduced by Grothendieck. The question of when these norms coincide leads to the concept of accessibility of tensor norms, and further on to injective and projective associate tensor norms that can be generated from a given tensor norm. The identification of the duals of the Chevet–Saphar norms leads to the ideals of \(p\)-integral operators. Further topics which are discussed in this chapter include \(2\)-factorable operators, the dual notion of \(2\)-dominated operators and, finally, Grothendieck’s list of fourteen natural tensor norms. The final Chapter relates general tensor norms to operator ideals in the sense of A. Pietsch.

Each chapter is accompanied by a set of (manageable) exercises. Moreover, there are three appendices. The second one deals with summability in Banach spaces, the third with spaces of measures, and the first one contains suggestions for further reading.

The book is very carefully written and edited. The text makes very good reading, in particular since no misprints seem to exist. Just one slip was noted: at least twice, A. Szankowski’s result that \(B(\ell_2)\) fails the approximation property is attributed to P. Enflo.

Grothendieck formulated his results using the language of tensor norms. Proper understanding of the involved concepts, however, requires much more than just some acquaintance. As a result, Grothendieck’s work, in particular his ‘théorème fondamental de la théorie métrique de produits tensoriels’, remained largely unnoticed for many years. This changed only in 1968 when J. Lindenstrauss and A. Pełczyński published their seminal paper [Stud. Math. 29, 275–326 (1968; Zbl 0183.40501)] in which they provided a matricial formulation of the ‘théorème fondamental’, nowadays known as Grothendieck’s inequality, and linked and extended many of Grothendieck’s results by using the theory of absolutely summing operators, as developed just one year earlier by A. Pietsch [Stud. Math. 28, 333–353 (1967; Zbl 0156.37903)]. In the sequel, the local theory of Banach spaces, for example, developed largely along these lines. However, up to few exceptions, the viewpoint of tensor norms was given up in favour of the easier accessible one of ideals of operators and their natural norms [cf. A. Pietsch’s monograph “Operator Ideals” (North–Holland Mathematical Library 20) (1980; Zbl 0434.47030)]. Several attempts have been undertaken to gain additional insight into topics like local Banach space theory by re-incorporating tensor norms into the developments. Most remarkably in this regard is the monograph by A. Defant and K. Floret [“Tensor Norms and Operator Ideals” (North–Holland Mathematics Studies 176) (1993; Zbl 0774.46018)].

Whereas the latter is a fairly advanced book, the one under review is intended to serve as an introduction to the theory of tensor products of Banach spaces. As such, it is a most welcome addition to the existing literature and appears to be well-suited as a guide and as a textbook in lectures, seminars, etc., for students having the standard facts from functional analysis and measure theory at their disposal.

The book starts by introducing, in the first chapter, tensor products of two vector spaces by means of ‘point evaluations’ on the corresponding space of bilinear forms; the basic algebraic facts are derived in a straightforward manner. Projective and injective tensor norms form the subject matter of the next two chapters, with emphasis on relations to Bochner and Pettis integrals, and to nuclear and integral operators, respectively. The fourth chapter deals with the approximation property and related matters. Among the topics which are discussed are reflexivity of spaces of operators, of tensor products, bases in tensor products of spaces with a basis, etc.

Chapter 5 is devoted to the Radon–Nikodym property for Banach spaces, to spaces of vector measures in comparison to projective and injective tensor products of spaces of scalar measures with the Banach space under discussion, and to the principle of local reflexivity. Chapter 6 starts with a brief introduction to general tensor norms. The main examples treated here are the Chevet–Saphar norms; it is explained how they lead to the ideals of \(p\)-summing operators. The standard results for such operators, including Grothendieck’s inequality, are derived. Chapter 7 compares Schatten’s ‘dual tensor norm’ with the corresponding one introduced by Grothendieck. The question of when these norms coincide leads to the concept of accessibility of tensor norms, and further on to injective and projective associate tensor norms that can be generated from a given tensor norm. The identification of the duals of the Chevet–Saphar norms leads to the ideals of \(p\)-integral operators. Further topics which are discussed in this chapter include \(2\)-factorable operators, the dual notion of \(2\)-dominated operators and, finally, Grothendieck’s list of fourteen natural tensor norms. The final Chapter relates general tensor norms to operator ideals in the sense of A. Pietsch.

Each chapter is accompanied by a set of (manageable) exercises. Moreover, there are three appendices. The second one deals with summability in Banach spaces, the third with spaces of measures, and the first one contains suggestions for further reading.

The book is very carefully written and edited. The text makes very good reading, in particular since no misprints seem to exist. Just one slip was noted: at least twice, A. Szankowski’s result that \(B(\ell_2)\) fails the approximation property is attributed to P. Enflo.

Reviewer: Hans Jarchow (Zürich)

### MathOverflow Questions:

Exposition of Grothendieck’s mathematicsProjective tensor product of injective operators

### MSC:

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |

46B28 | Spaces of operators; tensor products; approximation properties |

46M05 | Tensor products in functional analysis |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |