Topological Vector Spaces and Their Applications

This book gives modern coverage of locally convex topological vector spaces (LCTVS for short). The focus is on the spaces themselves rather than on how they are used.

There are lots of kinds of LCTVS and lots of kinds of properties they can have. This book concentrates on developing these, giving concrete examples, and showing which properties imply which other properties; it also includes many counterexamples for the cases where the implication is not valid. The book is marketed as a monograph and is probably too complex to be a textbook (although it does include a good collection of exercises).

The first two-thirds of the book develops the theory, concentrating on basic definitions, on construction of LCTVS, and on duality. This part of the book is a very focused study of the properties of LCTVS, without worrying too much about what they are used for. Duality plays a much larger role in this book than in some comparable books; the authors treat it primarily as a method of proof rather than a subject in itself. The other one-third of the book is the “Applications” of the title; it covers differentiable functions and integrals (that is, measures). The theory of derivatives in LCTVS has been greatly unified in recent years and this unified theory is presented in detail.

I think of derivatives and integrals not as applications but as additional properties of the spaces. There are also true applications scattered through the book; there’s a closing section in each chapter called “Complements” that deals with more specialized subjects that those in the body, and many of these are applications. For example, the inverse function theorem and the central limit theorem are covered in these sections. The body also has some applications, such as a treatment of the Fredholm integral equation.

Another book with a similar approach but broader coverage is Narici & Beckenstein’s Topological Vector Spaces. It takes a more general approach than the present book, as it deals with topological groups and general topological vector spaces, but its main interest is in LCTVS. Wilansky’s Modern Methods in Topological Vector Spaces is not as modern in approach but covers a lot of the same material. It’s not as intense a study, but it does provide a lot of the connections that will be needed for applications in analysis (without actually giving the applications). A good book going in the less-abstract direction is Hewitt & Stromberg’s Real and Abstract Analysis; this has very thorough coverage of integration and differentiation, but in Euclidean, Banach, and Hilbert spaces instead of general LCTVS. Rudin’s Functional Analysis is a classic work that deliberately avoids being encyclopedic but rather attempts to show the paths from general theories to other areas.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.