##
**Convergence structures and applications to functional analysis.**
*(English)*
Zbl 1246.46003

Berlin: Springer (ISBN 978-1-4020-0566-4/hbk; 978-90-481-5994-9/pbk). xiv, 264 p. (2002).

From the introduction: The class of convergence spaces is vast. It contains all topological spaces as well as many remarkable non-topological structures. Of particular interest here is the continuous convergence structure. This was introduced and studied for sequences by Hahn and Carathéodory and then studied in the general context of convergence spaces by Cook and Fischer as well as Binz and Keller. Perhaps the first major functional analytic result for convergence spaces was that of Butzmann: every complete locally convex topological vector space is ‘continuously’ reflexive. Since then, there has been much activity, both in generalizing classical functional analytic results to the larger convergence space setting and in analyzing the classical results with reference to a new duality.

There is much to be gained in such an analysis. Difficulties arise when one works exclusively with topological structures. First of all, topological vector spaces as well as topological groups lack a ‘natural’ dual structure. There is a whole spectrum of topologies on the dual of a topological vector space including, for example, the strong, Mackey, compact-open, weak, weak\(^*\dots\) to name a few. Each has its own advantages and difficulties. For example, the strong dual does not always distinguish between different topological vector spaces. Also, many of the best spaces of functional analysis, e.g, Fréchet spaces, may fail to be ‘strongly’ reflexive. Secondly, in topological vector spaces, sequences and series come into play all too seldom. Countability properties are always necessary before sequential arguments can be used and, in a topological vector space setting, these countability properties usually imply a Fréchet space. Finally, an inductive limit, a common and important object in functional analysis, seems very far removed from its component spaces. Consequently, properties of the component spaces are not easily preserved by the limit and properties of the limit are not easily lifted to the component spaces. An indication of the difficulty of these problems can be found in the survey paper by K.-D. Bierstedt [Lect. Int. Sch., Nice/Fr. 1986, 35–133 (1988; Zbl 0786.46001)].

The situation using convergence structures contrasts dramatically with the above situation. Continuous convergence provides a beautiful duality structure. Using the continuous dual and bidual, all complete locally convex topological vector spaces are ‘continuously’ reflexive. Also, countability properties are not nearly as restrictive for convergence vector spaces. For example, in Chapter 7 we shall see that, viewed as convergence vector spaces, the spaces of test functions and distributions are all second countable and sequential arguments suffice completely. Finally, in the setting of convergence vector spaces, inductive limits are much closer to their component spaces. As a result, properties of the components are much more easily preserved by the limit and properties of the limit much more easily lifted.

Renewed interest in Pontryagin’s duality theory for groups prompted an examination of group duality using continuous convergence in place of the compact open topology. Activity in extending the classical Pontryagin duality theory for topological groups has always been severely hindered by the fact that the canonical mapping of a topological group into its second character group fails, in general, to be continuous. We show that if, in place of the compact-open dual of a group one uses the continuous dual, one obtains a much improved duality theory.

It is the intention of this book to show that convergence vector spaces provide an excellent setting in which to do functional analysis, even if one’s interests do not go beyond topological groups or vector spaces. The convenience of convergence inductive limits, the ready access to countability properties and sequential arguments and, finally, the acquisition of structures like continuous convergence more than offset the problems of working in a much larger setting.

The authors had two purposes in writing this text. The first was to provide a systematic treatment of the theory of convergence spaces and, in particular, convergence vector spaces. The second was to illustrate how convergence structures, especially continuous convergence, can be useful in functional analysis.

Chapters 1 and 3 are introductions to convergence spaces and convergence vector spaces. The difficulties experienced by the authors in putting these chapters together is an indication of how scattered and anecdotal some of these results are. Chapter 2 contains a short introduction to uniform convergence space, the convergence generalization of uniform spaces. Although they do not enjoy all the strong properties of their uniform counterparts, many results resemble the classical ones. And, most importantly, uniform convergence spaces provide the right framework in which to prove an Arzelà-Ascoli theorem which generalizes both the classical uniform space version and the version used for convergence groups and vector spaces. It is hoped that these initial sections will prove to be a useful reference for others.

Chapters 4 through 8 deal with functional analytic applications of convergence groups and vector spaces. Chapter 4 analyzes continuous convergence as a duality structure. Of particular interest is the continuous dual of a locally convex topological vector space. Each of Chapters 5, 6 and 7 examines one particular fundamental theorem of functional analysis. The Hahn-Banach extension theorem, the closed graph theorem and the Banach-Steinhaus theorem are each examined in the vast setting of convergence vector spaces. We attempt to show how solutions to these problems for convergence vector spaces can be applied to problems of classical functional analysis. Chapter 8 deals with duality and reflexivity for convergence groups.

There is much to be gained in such an analysis. Difficulties arise when one works exclusively with topological structures. First of all, topological vector spaces as well as topological groups lack a ‘natural’ dual structure. There is a whole spectrum of topologies on the dual of a topological vector space including, for example, the strong, Mackey, compact-open, weak, weak\(^*\dots\) to name a few. Each has its own advantages and difficulties. For example, the strong dual does not always distinguish between different topological vector spaces. Also, many of the best spaces of functional analysis, e.g, Fréchet spaces, may fail to be ‘strongly’ reflexive. Secondly, in topological vector spaces, sequences and series come into play all too seldom. Countability properties are always necessary before sequential arguments can be used and, in a topological vector space setting, these countability properties usually imply a Fréchet space. Finally, an inductive limit, a common and important object in functional analysis, seems very far removed from its component spaces. Consequently, properties of the component spaces are not easily preserved by the limit and properties of the limit are not easily lifted to the component spaces. An indication of the difficulty of these problems can be found in the survey paper by K.-D. Bierstedt [Lect. Int. Sch., Nice/Fr. 1986, 35–133 (1988; Zbl 0786.46001)].

The situation using convergence structures contrasts dramatically with the above situation. Continuous convergence provides a beautiful duality structure. Using the continuous dual and bidual, all complete locally convex topological vector spaces are ‘continuously’ reflexive. Also, countability properties are not nearly as restrictive for convergence vector spaces. For example, in Chapter 7 we shall see that, viewed as convergence vector spaces, the spaces of test functions and distributions are all second countable and sequential arguments suffice completely. Finally, in the setting of convergence vector spaces, inductive limits are much closer to their component spaces. As a result, properties of the components are much more easily preserved by the limit and properties of the limit much more easily lifted.

Renewed interest in Pontryagin’s duality theory for groups prompted an examination of group duality using continuous convergence in place of the compact open topology. Activity in extending the classical Pontryagin duality theory for topological groups has always been severely hindered by the fact that the canonical mapping of a topological group into its second character group fails, in general, to be continuous. We show that if, in place of the compact-open dual of a group one uses the continuous dual, one obtains a much improved duality theory.

It is the intention of this book to show that convergence vector spaces provide an excellent setting in which to do functional analysis, even if one’s interests do not go beyond topological groups or vector spaces. The convenience of convergence inductive limits, the ready access to countability properties and sequential arguments and, finally, the acquisition of structures like continuous convergence more than offset the problems of working in a much larger setting.

The authors had two purposes in writing this text. The first was to provide a systematic treatment of the theory of convergence spaces and, in particular, convergence vector spaces. The second was to illustrate how convergence structures, especially continuous convergence, can be useful in functional analysis.

Chapters 1 and 3 are introductions to convergence spaces and convergence vector spaces. The difficulties experienced by the authors in putting these chapters together is an indication of how scattered and anecdotal some of these results are. Chapter 2 contains a short introduction to uniform convergence space, the convergence generalization of uniform spaces. Although they do not enjoy all the strong properties of their uniform counterparts, many results resemble the classical ones. And, most importantly, uniform convergence spaces provide the right framework in which to prove an Arzelà-Ascoli theorem which generalizes both the classical uniform space version and the version used for convergence groups and vector spaces. It is hoped that these initial sections will prove to be a useful reference for others.

Chapters 4 through 8 deal with functional analytic applications of convergence groups and vector spaces. Chapter 4 analyzes continuous convergence as a duality structure. Of particular interest is the continuous dual of a locally convex topological vector space. Each of Chapters 5, 6 and 7 examines one particular fundamental theorem of functional analysis. The Hahn-Banach extension theorem, the closed graph theorem and the Banach-Steinhaus theorem are each examined in the vast setting of convergence vector spaces. We attempt to show how solutions to these problems for convergence vector spaces can be applied to problems of classical functional analysis. Chapter 8 deals with duality and reflexivity for convergence groups.

### MSC:

46Axx | Topological linear spaces and related structures |

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