##
**Barrelled locally convex spaces.**
*(English)*
Zbl 0614.46001

North-Holland Mathematics Studies, 131. Notas de Matemática (113). Amsterdam etc.: North-Holland. XV, 512 p. $ 80.00; Dfl. 200.00 (1987).

From the authors’ introduction: “Our aim is to present a systematic treatment of barrelled spaces and of those structures in which barrelledness conditions are significant. We must advise the reader that this is not a book on applications of barrelled spaces to different areas of Functional Analysis, but a reasonably self-contained study of the structural theory of those spaces very much in the style of Köthe’s famous monographs. We have concentrated on presenting what we believe are basic phenomena in the theory and we have tried to display a variety of functional-analytic techniques.”

Contents: Introduction. Chapter 0 - Notations and preliminaries. Chapter 1 - Baire linear spaces. Chapter 2 - Basic tools. Chapter 3 - Barrels and disks. Chapter 4 - Barrelled spaces. Chapter 5 - Local completeness. Chapter 6 - Bornological and ultrabornological spaces. Chapter 7 - B- and \(B_ r\)-completeness. Chapter 8 - Inductive limit topologies. Chapter 9 - Strong barrelledness conditions. Chapter 10 - Locally convex properties of the space of continuous functions endowed with the compact-open topology. Chapter 11 - Barrelledness conditions on topological tensor products. Chapter 12 - Holomorphically significant properties of locally convex spaces. Chapter 13 - A short collection of open problems. A table of barrelled spaces. References. Tables. Index.

In their aims, the authors succeed remarkably well. They complement the ”usual” discussion of barrelled and bornological spaces (as well as the introductory theory of locally convex inductive limits) which can be found in the well-known books on topological vector spaces by presenting a wealth of material (some of it very recent) which had not appeared in book form before. The attentive reader can appreciate how much work (and “expertise”) went into the preparation of this monograph.

Barrelled and bornological spaces were introduced in connection with the generalization of some of the classical theorems in Banach or Fréchet spaces to the setting of locally convex (l.c.) spaces, and a large part of the present book is devoted to aspects of closed graph, open mapping and uniform boundedness theorems. The authors study Baire linear spaces and L. Schwartz’s “borelian graph theorem”, \(B\)- and \(B_ r\)-completeness and Pták’s theory, but also aspects of de Wilde’s approach and the recent conributions of M. Valdivia in this area. Basic tools like the sliding hump technique and normed spaces \(E_ B\) associated with a disk \(B\) in a l.c. space \(E\) are discussed to some extent, and real “connaisseurs” will find many interesting methods and nice (technical) details by inspecting the proofs of some of the harder and deeper theorems in this book. Some parts may really only be appreciated by researchers interested in details in the structure theory of l.c. spaces (others will sometimes question why so many different, but closely related classes of spaces are introduced, like e.g. \(G\)-barrelled, suprabarrelled, totally barrelled, Baire-like, quasi-Baire spaces etc.). But Fréchet spaces and their duals play a very prominent role throughout the text, and the chapters on inductive limit topologies and topological tensor products certainly are major highlights!

While the book really is reasonably self-contained, its appreciation probability presupposes the knowledge of large parts of the theory of topological vector spaces and of some of its applications. And while applications are not in the center of the monograph, the treatment of topics like \((DF)\)- and \((gDF)\)-spaces, l.c. properties of the spaces \(C(X)\) and \(C_ c(X)\), weighted inductive limits and their projective descriptions, tensor products of Fréchet and \((DF)\)-spaces (including recent work of D. Vogt), as well as holomorphically significant properties of l.c. spaces is also very welcome from the point of view of the applications.

Another welcome feature are the sections “Notes and remarks” at the ends of the chapters, where credit for the results is given, but also further results are discussed and outlined, sometimes with full proofs.

Some of the chapters follow the pattern “definitions-characterizations- permanence properties-examples and counterexamples”, and there the style is somewhat terse. On the other hand, the authors often add real “gems” in their “Notes and remarks” sections, e.g. 3.4.3 and 3.4.4 are taken from S. Grabiner’s [Am. Math. Mon. 93, 190–191 (1986; Zbl 0637.46004)] article, 11.10.C presents an easy proof (due to Kaijser) of Grothendieck’s inequality, and 11.10.D gives Ransford’s recent proof of (Machado’s version of) Bishop’s Stone-Weierstrass theorem.

There is a reasonable number of misprints (mostly obvious). (In 0.1.1.(c), “locally compact” should be replaced by “Lindelöf”, and the reference to quasinormable spaces in the index should read “8.3.34”.) Some of the most interesting open problems in Chapter 13 were solved by J. Taskinen just during the last months. There is only a limited amount of overlap with, say, parts of the second volume of Köthe’s monograph and of the monograph of H. Jarchow. Of course, the present book only deals with one part of “what is going on” in topological vector spaces, and, in fact, in some other areas important research is being done. But, from this reviewer’s point of view, the book of Pérez Carreras-Bonet can indeed be recommended warmly to anybody interested in the modern theory of locally convex spaces.

Contents: Introduction. Chapter 0 - Notations and preliminaries. Chapter 1 - Baire linear spaces. Chapter 2 - Basic tools. Chapter 3 - Barrels and disks. Chapter 4 - Barrelled spaces. Chapter 5 - Local completeness. Chapter 6 - Bornological and ultrabornological spaces. Chapter 7 - B- and \(B_ r\)-completeness. Chapter 8 - Inductive limit topologies. Chapter 9 - Strong barrelledness conditions. Chapter 10 - Locally convex properties of the space of continuous functions endowed with the compact-open topology. Chapter 11 - Barrelledness conditions on topological tensor products. Chapter 12 - Holomorphically significant properties of locally convex spaces. Chapter 13 - A short collection of open problems. A table of barrelled spaces. References. Tables. Index.

In their aims, the authors succeed remarkably well. They complement the ”usual” discussion of barrelled and bornological spaces (as well as the introductory theory of locally convex inductive limits) which can be found in the well-known books on topological vector spaces by presenting a wealth of material (some of it very recent) which had not appeared in book form before. The attentive reader can appreciate how much work (and “expertise”) went into the preparation of this monograph.

Barrelled and bornological spaces were introduced in connection with the generalization of some of the classical theorems in Banach or Fréchet spaces to the setting of locally convex (l.c.) spaces, and a large part of the present book is devoted to aspects of closed graph, open mapping and uniform boundedness theorems. The authors study Baire linear spaces and L. Schwartz’s “borelian graph theorem”, \(B\)- and \(B_ r\)-completeness and Pták’s theory, but also aspects of de Wilde’s approach and the recent conributions of M. Valdivia in this area. Basic tools like the sliding hump technique and normed spaces \(E_ B\) associated with a disk \(B\) in a l.c. space \(E\) are discussed to some extent, and real “connaisseurs” will find many interesting methods and nice (technical) details by inspecting the proofs of some of the harder and deeper theorems in this book. Some parts may really only be appreciated by researchers interested in details in the structure theory of l.c. spaces (others will sometimes question why so many different, but closely related classes of spaces are introduced, like e.g. \(G\)-barrelled, suprabarrelled, totally barrelled, Baire-like, quasi-Baire spaces etc.). But Fréchet spaces and their duals play a very prominent role throughout the text, and the chapters on inductive limit topologies and topological tensor products certainly are major highlights!

While the book really is reasonably self-contained, its appreciation probability presupposes the knowledge of large parts of the theory of topological vector spaces and of some of its applications. And while applications are not in the center of the monograph, the treatment of topics like \((DF)\)- and \((gDF)\)-spaces, l.c. properties of the spaces \(C(X)\) and \(C_ c(X)\), weighted inductive limits and their projective descriptions, tensor products of Fréchet and \((DF)\)-spaces (including recent work of D. Vogt), as well as holomorphically significant properties of l.c. spaces is also very welcome from the point of view of the applications.

Another welcome feature are the sections “Notes and remarks” at the ends of the chapters, where credit for the results is given, but also further results are discussed and outlined, sometimes with full proofs.

Some of the chapters follow the pattern “definitions-characterizations- permanence properties-examples and counterexamples”, and there the style is somewhat terse. On the other hand, the authors often add real “gems” in their “Notes and remarks” sections, e.g. 3.4.3 and 3.4.4 are taken from S. Grabiner’s [Am. Math. Mon. 93, 190–191 (1986; Zbl 0637.46004)] article, 11.10.C presents an easy proof (due to Kaijser) of Grothendieck’s inequality, and 11.10.D gives Ransford’s recent proof of (Machado’s version of) Bishop’s Stone-Weierstrass theorem.

There is a reasonable number of misprints (mostly obvious). (In 0.1.1.(c), “locally compact” should be replaced by “Lindelöf”, and the reference to quasinormable spaces in the index should read “8.3.34”.) Some of the most interesting open problems in Chapter 13 were solved by J. Taskinen just during the last months. There is only a limited amount of overlap with, say, parts of the second volume of Köthe’s monograph and of the monograph of H. Jarchow. Of course, the present book only deals with one part of “what is going on” in topological vector spaces, and, in fact, in some other areas important research is being done. But, from this reviewer’s point of view, the book of Pérez Carreras-Bonet can indeed be recommended warmly to anybody interested in the modern theory of locally convex spaces.

Reviewer: Klaus Dieter Bierstedt (Paderborn)

### MSC:

46A08 | Barrelled spaces, bornological spaces |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46A30 | Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness) |

46G20 | Infinite-dimensional holomorphy |

46M05 | Tensor products in functional analysis |

46F05 | Topological linear spaces of test functions, distributions and ultradistributions |

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

46M40 | Inductive and projective limits in functional analysis |