This text is designed as a basic course in functional analysis for senior undergraduate or beginning postgraduate students. For students completing their final undergraduate year, it is aimed at providing some insight into basic abstract analysis which more than ever, is the contextual language of much modern mathematics. For postgraduate students it is aimed at providing a foundation and stimulus for their further research development.

It is assumed that the student will be familiar with real analysis and have some background in linear algebra and complex analysis. It is also assumed that the student will have studied a course in the analysis of metric spaces such as that given in the author's text

Introduction to the Analysis of Metric Spaces, Cambridge University Press, 1987. Reference to this text will be made under the abbreviation AMS §—.

In AMS, most of the example spaces introduced are normed linear spaces and many of the implications of linear structure were explored. For example when closure in metric spaces was discussed it was natural to consider the closure of linear subspaces in normed linear spaces and when continuity was considered it was logical to study the continuity of linear mappings on normed linear spaces. In order to make this text as self-contained as possible, the example spaces are again introduced and the elementary properties of normed linear spaces are treated but in a more sophisticated way.

Extension theory for symmetric operators

Extension theory and Nevanlinna R-functions

Probably the most interesting example zof a singular rank one perturbation is the perturbation of the boundary condition for a second order ordinary differential operator. This is one of the first mathematical problems where the extension theory plays an indispensable role. In 1909–1910 H. Weyl investigated in his famous papers [954, 955] the behavior of the solutions to the second order differential equation under variation of the boundary condition. He was also the first to ask the question: How does the spectrum change under such a perturbation? He proved that the absolutely continuous spectrum is invariant under such a perturbation. The question by H.Weyl concerning other types of spectrum has been investigated by F. Wolf [964], N. Aronszajn [97], and N.Aronszajn and W.F.Donoghue [99]. See also the paper by V.A. Javrian [493]. So it was H.Weyl who was the first to understand the importance of this class of perturbations from the mathematical point of view. The first mathematically rigorous investigation of singular perturbations of partial differential operator was carried out by F.A.Berezin and L.D.Faddeev [135]. These authors have shown that such perturbations can be described using the extension theory of symmetric operators. This paper was extremely important because it clarified the relation between partial differential operators with point interactions and Krein's formula describing the resolvents of all self–adjoint extensions of a given symmetric operator.

In this monograph we study systematically certain classes of perturbations of given self–adjoint operators in Hilbert spaces. As main examples we consider second order differential operators in L2-spaces perturbed by finite or infinite rank operators, respectively by certain generalized ‘interaction terms’. Typical results concern spectral properties and scattering quantities. The operators we discuss include as special cases Hamiltonians with ‘point interactions’, i.e., interactions involving potentials of the δ, respectively δ′-type supported by a (finite or infinite) set of isolated points or suitable lower dimensional Irypersurfaces. Such Hamiltonians occur, e.g., in the description of quantum mechanical systems in solid state physics, atomic and nuclear physics as well as in the description of electromagnetic phenomena, in the modelling of certain related chemical and biological phenomena, and in the study of quantum chaotic systems.

Concerning these ‘point interaction models’, in the last decade two specific monographs have appeared along with a few proceedings, books and specialized papers. One of the main aims of the present book is to present a natural continuation of the previous work [39], much in the same rigorous mathematical spirit, and covering some of the developments which occurred after the appearance of [39] (and its Russian improved version [40]). Our present book extends the analysis of [39] (and [40]) in two directions. On one hand we look at the operators discussed in [39] as special cases of a general theory of (singular) perturbations of (differential) operators.

In addition to the Hahn–Banach Theorem the three mapping theorems, the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem are vital for the development of any general theory of Banach spaces.

In these theorems we begin to appreciate the importance of the completeness condition. The proofs are based on Baire category arguments which reveal the implications of completeness for the metric topology. So we begin by developing this theory and demonstrate something of its force before applying it to establish the fundamental mapping theorems.

BAIRE CATEGORY THEORY FOR METRIC SPACES

In complete metric spaces the metric topology has important characteristics and a knowledge of these is indispensable in establishing many significant results in the analysis of normed linear spaces and elsewhere.

We recall the following definition from the analysis of metric spaces.

Definition. Given a metric space (X, d), a subset A is said to be dense in (X, d) if its closure Ā = X.

This means that A is dense in (X, d) if and only if every point of X is either a point of A or a cluster point of A. Equivalently, A is dense in (X, d) if and only if for every x ∈ X and ε > 0, we have B(x; ε) ∩ A ≠ ∅.

The following concept related to density is used to partition metric spaces into disjoint classes.