Outline of Functional Analysis

Abstract

Problems in PDE have provided a major impetus for the development of functional analysis. Here, we present some basic results, which are useful for the development of such subjects as distribution theory and Sobolev spaces, discussed in Chaps. 3 and 4; the spectral theory of compact and of unbounded operators, applied to elliptic PDE in Chap. 5; the theory of Fredholm operators and their indices, needed for the study of the Atiyah–Singer index theorem in Chap. 10; and the theory of semigroups, of particular value in Chap. 9 on scattering theory, and also germane to studies of evolution equations in Chaps. 3 and 6. Indeed, what is thought of as the subject of functional analysis naturally encompasses some of the development of these chapters as well as the material presented in this appendix. One particular case of this is the spectral theory of Chap. 8. In fact, it is there that we present a proof of the spectral theorem for general self-adjoint operators. One reason for choosing to do it this way is that my favorite approach to the spectral theorem uses Fourier analysis, which is not applied in this appendix, though some of the exercises make contact with it. Thus in this appendix the spectral theorem is proved only for compact operators, an extremely simple special case. On the other hand, it is hoped that by the time one gets through the Fourier analysis as developed in Chap. 3, the presentation of the general spectral theorem in Chap. 8 will appear to be very simple too.