Modern Methods in Topological Vector Spaces

A topological vector space is a vector space equipped with a topology such that the vector operations of addition and scalar multiplication are continuous. The general idea of this theory is to see how much of the theory of normed linear spaces can be proven without the hypothesis of a norm, and in particular how the concept of bounded set can be generalized. The theory goes back to von Neumann in the 1930s with his introduction of locally convex spaces. To some extent it is a generalization of functional analysis to non-normable spaces, and a lot of the same theorems, such as Hahn–Banach, appear in both theories. This more general theory is useful in the theory of distributions, and in a variety of function spaces such as infinitely differentiable functions and in the study of uniform convergence on compact subsets.

The present book is presented as a text for a one-year graduate course, but I think this doesn’t work. The number of kinds of spaces is overwhelming, and no one would be able to remember more than a few of them after a year. The book reads more like a catalog than a text, and there isn’t a good guide to which spaces and theorems are the most important. It’s also very skimpy on examples and applications, although it has a ridiculous number of exercises (claimed to be 1500 by the author; these are divided into four levels of difficulty). It works better as a reference, especially because there is a very helpful table of all the spaces in the back, showing their relationships and which ones imply which other ones.

One bad feature, in a book with so many definitions, is that many of them are not explicit but are buried in the exercises or in the statements of the theorems. Happily the index seems to be complete and you can always find your definition using it.

The present work is a Dover unaltered reprint from 2013 of the 1978 McGraw-Hill edition, and it’s natural to wonder if the title word “Modern” still applies. I think it does, although there has been a shift away from duality, which is the key concept in the present book. A more recent book, with the same coverage, is Narici & Beckenstein’s Topological Vector Spaces (first edition 1985, second edition 2010). The authors explicitly state that their book is intended as a reference and not to be read straight through.

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