We discuss in detail p-absolutely summing operators. The Pietsch factorization theorem, which is basic to this theory, is proved. The fundamental Grothendieck theorem is proved in its three most useful forms. Later we improve it and show the Grothendieck-Maurey theorem, that every operator from any L1-space into a Hilbert space is p-absolutely summing for all p > 0. We present the trace duality and show that the p′-nuclear norm is dual to the p-absolutely summing norm. We also introduce and discuss p-integral operators. We show the connection between cotype 2 and the coincidence of classes of p-absolutely summing operators for various p's. The extrapolation result for p-absolutely summing operators is proved. We apply Grothendieck's theorem to exhibit examples of power bounded but not polynomially bounded operators on a Hilbert space and to give some estimates for the norm of a polynomial of a power-bounded operator. We also present many applications to harmonic analysis: we construct good local units in L1(G), we prove the classical Orlicz-Paley-Sidon theorem and give some characterizations of Sidon sets.

1. In this chapter we will discuss several important classes of operators, namely p-absolutely summing, p-integral and p-nuclear operators. All these classes have some ideal properties so we will introduce the general concept of an operator ideal.

The sets compact in the σ(X, X*)-topology are important in many applications. We study such sets in this section. The main result is the Eberlein-Smulian theorem which says that weak compactness of a set is determined by properties of sequences, even when the σ(X, X*)-topology on this set is not metrizable. We apply this to study weakly compact operators, i.e. operators such that the image of any ball is contained in a weakly compact set. We show that each weakly compact operator factorizes through a reflexive space, and use this to investigate properties of such operators.

1. This section is devoted to the study of weakly compact sets in Banach spaces, i.e. subsets A ⊂ X which are compact in the σ(X, X*)-topology. We say that the set A ⊂ X is relatively weakly compact if its σ(X, X*)-closure in X is weakly compact. Prom Theorem II.A. 14 we infer that every bounded subset of a reflexive space is relatively weakly compact. Also by Theorem II.A.4 and II.A. 14 we get that every convex, bounded, norm-closed subset of a reflexive space is weakly compact. Also if X is a reflexive space and if T: X → Y is a continuous linear operator, then T(Bx) is a weakly compact set.

2. We have

Lemma.A subset A ⊂ X is relatively weakly compact if and only if it is bounded and the σ(X**, X*)-ciosure of i(A) in X** is contained in i(X).

In this chapter we present some results and notions concerning finite dimensional Banach spaces and the relation between an infinite dimensional Banach space and its finite dimensional subspaces. We start with a discussion of the bounded approximation property and the TTA-spaces. We also prove the local reflexivity principle which connects the local properties of X and X**. We prove the Auerbach lemma which allows a good identification of an n-dimensional Banach space with ℝn or ℂn. We also study the concept of Banach-Mazur distance.

1. By local properties of a Banach space we mean the properties which depend on the structure of finite dimensional subspaces of the space. Some examples of such properties will be pointed out in this chapter and many more will be encountered in the sequel.

The basic aim of this chapter is to provide an elementary understanding of local phenomena. Even at this early stage it is apparent that one needs a clarification of two points:

1. (a) how the general Banach space is built up from finite dimensional subspaces;

2. (b) what are the relevant properties of finite dimensional spaces.

Let us start with some definitions and examples which explain point (a) a little. What we are really thinking about in (a) is the approximation problem: how well can we approximate the identity operator on the space X by finite dimensional operators?

1. A linear topological space X is a linear space over the real or complex numbers endowed with a topology τ such that the map (x, y) ↦ x + y is continuous from (X, τ) × (X, τ) into (X, τ) and the map (t, x) ↦ tx is continuous from ℝ × X (or ℂ × X) into X. Such a topology is fully described by a basis of neighbourhoods of 0. A subset V ⊂ X is called convex if whenever x1, x2 ∈ V then the whole interval αx1 + (1 – α)x2 for 0 ≤ α ≤ 1 is in V. A linear topological space is called locally convex if it has a basis of convex neighbourhoods of 0. A functional on X is a continuous linear map from X into scalars. The set of all functionals on X will be denoted X*, and called the dual space. A linear operator (or just operator) T : X → Y (where X and Y are linear topological spaces) is a continuous linear map. A subspace of X will always (unless explicitly stated otherwise) denote a closed linear subspace. Given a set V ⊂ X by spanV we denote the closure of the set of all linear combinations of elements from V (i.e. the subspace of X spanned by V).

2. A linear topological space X is called an F-space if its topology is given by a metric ρ such that ρ(x, y) = ρ(x - y, 0) and X is complete with respect to this metric.

This introductory part contains background material. It is not intended to be a course in any subject. It is simply a collection of definitions and facts given without proof (with one exception). We provide references to works which contain detailed exposition, full proofs, examples and motivation. In a sense this whole part is a quick reference guide to results which will be used in the later parts. The introduction is divided into two chapters, Chapter I.A describing what we will need from general functional analysis and Chapter I.B which contains results about concrete spaces and operators. Since I.A is really a review of a standard course in functional analysis, references are given only at the end of the chapter. In I.B we give references after each paragraph.

The references given in this part are usually to standard textbooks and monographs, not to original works. If a particular result or subject cannot be easily located using the table of contents or index we try to provide more detailed information (sections or pages). Sometimes we formulate a result in a form which is more convenient to us but different from the one given in the reference. Usually in such a case it is easy to derive our formulation from the one given in the reference.