Entropy and compactness properties of operators are closely related to approximation properties. ‘Approximation’ means approximation by finite rank operators. Approximation quantities are non-increasing sequences of non-negative numbers sn(T) defined for arbitrary operators T between Banach spaces and, in some sense, express the degree of approximability of T by finite rank operators. We shall deal with the so-called approximation numbers and with Kolmogorov and Gelfand numbers. In their original meaning both Kolmogorov and Gelfand numbers – like entropy numbers – are set functions. They entered the mathematical literature as certain diameters of sets. The definitive paper by Kolmogorov appeared in 1936 (cf. Kolmogorov 1936). For the diameters in the sense of Gelfand, we do not feel able to date their origin. However, for a detailed and comprehensive survey of the development of the theory of diameters we recommend the book by Pinkus (1985). In the present book, for the sake of economy, we shall confine ourselves to Kolmogorov and Gelfand numbers of operators as they were considered within a general theory of so-called s-numbers of operators by Pietsch (see Pietsch 1974, 1978). Nevertheless, we refer to the geometrical meaning of Kolmogorov numbers and Gelfand numbers. The particular definition of Gelfand numbers that we give (see (2.3.5)) has not been used in literature so far (see Stephani 1987). It is thought to emphasize the analogy to the geometrical definition of the Kolmogorov numbers (see (2.2.4)) irrespective of duality arguments.

This book deals with a branch of modern functional analysis which has arisen only in the last 10 years, although it has its origin in a 1932 paper by Pontrjagin and Schnirelman.

In general there is quite a big difference between the level of recent research and the level of lectures as they are given to students. The question arises if this is in the nature of the subject, or if it is mainly a problem of producing an appropriate representation of the subject. Concerning ‘Entropy, compactness and the approximation of operators’, we came to the opinion that it should be possible to represent the subject at a level which makes reference only to the results of an introductory course on functional analysis. We have tried to write the book in the corresponding style and have listed in the introduction the concepts necessary for an understanding of the book. A few facts beyond the standard elementary knowledge of functional analysis are used without proof. However, a reader who is only interested in the fundamental relations between entropy quantities, approximation quantities, and eigenvalues can leave out the more difficult passages. By reading only sections 1.1, 1.2, 1.3, 1.4 of chapter 1, section 2.1 of chapter 2, section 3.1 of chapter 3, and section 4.2 of chapter 4, he or she will get an impression of the main ideas of the book and will be able to follow the applications of the general results in chapter 5.

We have been using the term ‘fixed point property’ (f.p.p.) as it relates to the class of nonexpansive mappings but, of course, this property may be applied to any class of mappings. The ‘topological’ fixed point property is of fundamental importance in the broader context of fixed point theory.

Definition 18.1 A topological space X is said to have the (topological) fixed point property (t.f.p.p. or f.p.p. if the meaning is clear) if each continuous mapping T: X→X has a fixed point.

It is not surprising that the above property is a topological invariant.

Lemma 18.1 If X and Y are homeomorphic and if X has the t.f.p.p., then Y also has the t.f.p.p.

Proof Let h: X→Y be a homeomorphism with h(X) = Y, and suppose f: Y→Y is continuous. Then g = h–1 ∘ f ∘ h: X→X and g is continuous; hence there exists x ∈ X such that gx = x, implying y = hx = fy.

Another very useful observation about the t.f.p.p. is the following:

Lemma 18.2 If a topological space X has the t.f.p.p. and if Y is a retract of X, then Y has the t.f.p.p.

Proof. Let r be a retraction of X onto Y and let f: Y→Y be continuous. Then f may be extended to a continuous mapping g = f ∘ r: X→X. Any fixed point of g must also be a fixed point of f.