Preface

About functional analysis. There is a common opinion that algebra studies sets endowed with operations, such as, say, addition, multiplication, taking the inverse or symmetric element. On the other hand, topology studies sets where continuous passing from some elements to others makes sense, and in particular, where convergent sequences are defined. With the same oversimplification, we can say that functional analysis is the study of sets with synthetic (i.e., composite) structure, partly algebraic and partly topological, and the two parts are related to each other by certain natural rules. In the original form of classical functional analysis of the 1920s the algebraic structure is the linear space structure, while the convergence of vectors is defined (and made compatible with algebraic operations) via a given norm. However, it was immediately discovered that for the main part the really deep results require the additional assumption of completeness of the normed space under consideration. This led to the notion of Banach space. But the greatest progress in classical functional analysis was achieved in the theory of Hilbert spaces, which are defined as Banach spaces with norm given by an inner product. Here “complete success” was achieved in a number of fundamental directions. Notably, we now know everything about the nature of Hilbert spaces (the Riesz–Fisher theorem) and about the most important classes of maps of these spaces (the Hilbert spectral theorem, the Schmidt theorem). (The classical functional analysis of normed, Banach, and Hilbert spaces constitutes the main part of our book.) As time passed, new problems required enrichment of the initial structures. First of all, it was realized that to study many important function