Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra

Abstract

The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system inevitably tends to a disappointment. “Do not seek any other patterns or connections but those that we have described here, because any of your assumptions will find a counterexample in Nature”; this phrase should have been cited as an afterward to the three textbooks on topological vector spaces translated into Russian [12, 41, 43], which represent the face of this science from the sixties until now. Indeed, after the famous Enflo counterexample [22], the development of the theory of topological vector spaces (which was planned originally as an extension of a clear and simple discipline, linear algebra) was transformed into a sequence of reports on the oddities one can encounter in modern mathematics. The result of this process is a general skepticism about and a loss of interest in this field. The number of papers on the theory of TVS is declining, while the attitude of other mathematicians now consists of polite perplexity or lenient irony. And add to that the fact that the vast majority of specialists in functional analysis do not concern themselves with topological vector spaces but consciously concentrate on Banach theory. (According to Compumath, for example, in 1996 only 7 papers were devoted to locally convex and topological vector spaces, while 612 were devoted to Banach spaces. In the theory of topological algebras, the ratio is 6:90.) The dominant position of Banach theory in functional analysis justifies a natural question: what are the advantages of the Banach case over the more general topological situation? At the conceptual level, the difference becomes most apparent in the theory of topological algebras. Let us cast a critical look. The first impression we will get when comparing Banach and non-Banach theories of topological algebras is as follows. While the concept of Banach algebra emerges naturally from intuitive expectations, faciliated by an acquaintance with general algebra and the theory of Banach spaces (and leads to the profound theory of Banach algebras), the situation with general topological algebras looks completely different. We find here unexpectedly that the very attempts to define a topological algebra (and topological module) lead to results that contradict intuition. Indeed, intuitively it is clear that a “good” definition of topological algebra (and topological module) must fulfill at least the following minimal list of requirements: