3 - Topological Vector Spaces

Abstract

In the 1930s and 1940s, after the publication of the book by S. Banach, the mathematical community might have been forgiven for thinking that functional analysis would henceforth essentially be set in stone, given the natural framework provided by normed vector spaces (and their completions: Banach spaces). But the holomorphic functions on an open set in the complex plane are just one example of a topogical vector space that is non-normable despite having the structure of a Fréchet-Montel space. The general theory of topological vector spaces was outlined by A. Kolmogorov and J. von Neumann in 1935, then completed in 1945-1946 by the fundamental contributions of G. Mackey on locally convex spaces. The significance of these generalizations became clear with the advent of distribution theory, developed by L. Schwartz between 1945 and 1950 following precursory work by S. Sobolev and S. Bochner; this new theory propelled a previously unheard of type of topological vector space into the spotlight. Dieudonné and Schwartz showed in an article published in 1949 that these spaces are in fact strict inductive limits of Fréchet spaces. The groundwork for this article had been laid by a summary published in 1942 by Dieudonné on duality in locally convex spaces, as well as the contributions by Mackey mentioned above. The article by Dieudonné-Schwartz was completed in the early 1950s by A. Grothendieck: generalization of the open mapping theorem and the closed graph theorem, notions of a Schwartz space and of a nuclear space, and general theory of tensor products of topological vector spaces (which we sadly do not have the space to discuss here beyond a few basic ideas). Other generalizations of the open mapping theorem and the closed graph theorem can also be found in the literature, notably the ones derived by V. Pták. For an overview of these results, readers can refer to (Volume II).