In the previous chapter we saw that for every compact operator T on a Banach space X, the space can almost be written as a direct sum of generalized eigenspaces of T. If we assume that X is not merely a Banach space, but a Hilbert space, and T is not only compact but compact and normal, then such a decomposition is indeed possible – in fact, there is a decomposition with even better properties. Such a decomposition will be provided by the spectral theorem for compact normal operators: a complete and very simple description of compact normal operators. Thus with the study of a compact normal operator on a Hilbert space we arrive in the promised land: everything fits, everything works out beautifully, there are no blemishes. This is the best of all possible worlds.

We shall give two proofs of the spectral theorem, claiming the existence of the desired decomposition. In the first proof we shall make use of some substantial results from previous chapters, including one of the important results concerning the spectrum of a compact operator. The second proof is self-contained: we shall replace the results of the earlier chapters by easier direct arguments concerning Hilbert spaces and normal operators.

To start with, we collect a number of basic facts concerning normal operators in the following lemma. Most of these facts have already been proved, but for the sake of convenience we prove them again.

Lemma 1. Let T ∈ B(H) be a normal operator. Then the following assertions hold.

Given a complex Banach space X, which operators T ∈ B(X) have nontrivial closed invariant subspaces? This question, the so-called invariant subspace problem, is the topic of this brief last chapter. Until fairly recently, it was not known whether there was any operator T without a non-trivial (closed) invariant subspace, and it is still not known whether there is such an operator on a (complex) Hilbert space.

Much of the effort concerning the invariant-subspace problem has gone into proving positive results, i.e. results claiming the existence of invariant subspaces for operators satisfying certain conditions. Our main aim in this chapter is to present the most beautiful of these positive results, Lomonosov's theorem, whose proof is surprisingly simple.

As we remarked earlier, the Riesz theory of compact operators on Banach spaces culminated in a very pleasing theorem, Theorem 13.8, which nevertheless, did not even guarantee the existence of a single nontrivial invariant subspace. This deficiency was put right, with plenty to spare, in Chapter 14, but only for a compact normal operator on a Hilbert space. Now we return to the general case to prove Lomonosov's theorem, which claims considerably more than that every compact operator has a non-trivial invariant subspace. Before we present this result, we need some definitions and a basic result about compact convex sets.

As in Chapters 13 and 14, all spaces considered in this chapter are complex spaces. Furthermore, as every linear operator on a finitedimensional complex vector space has an eigenvector, we shall consider only infinite-dimensional spaces.

This book has grown out of the Linear Analysis course given in Cambridge on numerous occasions for the third-year undergraduates reading mathematics. It is intended to be a fairly concise, yet readable and down-to-earth, introduction to functional analysis, with plenty of challenging exercises. In common with many authors, I have tried to write the kind of book that I would have liked to have learned from as an undergraduate. I am convinced that functional analysis is a particularly beautiful and elegant area of mathematics, and I have tried to convey my enthusiasm to the reader.

In most universities, the courses covering the contents of this book are given under the heading of Functional Analysis; the name Linear Analysis has been chosen to emphasize that most of the material in on linear functional analysis. Functional Analysis, in its wide sense, includes partial differential equations, stochastic theory and non-commutative harmonic analysis, but its core is the study of normed spaces, together with linear functionals and operators on them. That core is the principal topic of this volume.

Functional analysis was born around the turn of the century, and within a few years, after an amazing burst of development, it was a wellestablished major branch of mathematics. The early growth of functional analysis was based on 19th century Italian function theory, and was given a great impetus by the birth of Lebesgue's theory of integration.