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Über dieses Buch

This work grew out of Errett Bishop's fundamental treatise 'Founda­ tions of Constructive Analysis' (FCA), which appeared in 1967 and which contained the bountiful harvest of a remarkably short period of research by its author. Truly, FCA was an exceptional book, not only because of the quantity of original material it contained, but also as a demonstration of the practicability of a program which most ma­ thematicians believed impossible to carry out. Errett's book went out of print shortly after its publication, and no second edition was produced by its publishers. Some years later, 'by a set of curious chances', it was agreed that a new edition of FCA would be published by Springer Verlag, the revision being carried out by me under Errett's supervision; at the same time, Errett gener­ ously insisted that I become a joint author. The revision turned out to be much more substantial than we had anticipated, and took longer than we would have wished. Indeed, tragically, Errett died before the work was completed. The present book is the result of our efforts. Although substantially based on FCA, it contains so much new material, and such full revision and expansion of the old, that it is essentially a new book. For this reason, and also to preserve the integrity of the original, I decided to give our joint work a title of its own. Most of the new material outside Chapter 5 originated with Errett.

Inhaltsverzeichnis

Frontmatter

Prolog

Abstract
Most mathematicians would find it hard to believe that there could be any serious controversy about the foundations of mathematics, any controversy whose outcome could significantly affect their own mathematical activity. Their attitude well represents the actual state of affairs: during a half-century of splendid mathematical progress there has been no deviation from the norm. The voices of dissent, never much heeded, have long been silent.
Errett Bishop, Douglas Bridges

Chapter 1. A Constructivist Manifesto

Abstract
Mathematics is that portion of our intellectual activity which transcends our biology and our environment. The principles of biology as we know them may apply to life forms on other worlds, yet there is no necessity for this to be so. The principles of physics should be more universal, yet it is easy to imagine another universe governed by different physical laws. Mathematics, a creation of mind, is less arbitrary than biology or physics, creations of nature; the creatures we imagine inhabiting another world in another universe, with another biology and another physics, will develop a mathematics which in essence is the same as ours. In believing this we may be falling into a trap: Mathematics being a creation of our mind, it is, of course, difficult to imagine how mathematics could be other than it is without our actually making it so, but perhaps we should not presume to predict the course of the mathematical activities of all possible types of intelligence. On the other hand, the pragmatic content of our belief in the transcendence of mathematics has nothing to do with alien forms of life. Rather, it serves to give a direction to mathematical investigation, resulting from the insistence that mathematics be born of an inner necessity.
Errett Bishop, Douglas Bridges

Chapter 2. Calculus and the Real Numbers

Abstract
Section 1 establishes some conventions about sets and functions. The next three sections are devoted to constructing the real numbers as certain Cauchy sequences of rational numbers, and investigating their order and arithmetic. The rest of the chapter deals with the basic ideas of the calculus of one variable. Topics covered include continuity, the convergence of sequences and series of continuous functions, differentiation, integration, Taylor’s theorem, and the basic properties of the exponential and trigonometric functions and their inverses. Most of the material is a routine constructivization of the corresponding part of classical mathematics; for this reason it affords a good introduction to the constructive approach.
Errett Bishop, Douglas Bridges

Chapter 3. Set Theory

Abstract
The chapter begins with a fuller discussion of sets and functions, and includes the constructive meaning of such terms as subset, union, and inclusion. Certain classical laws of the algebra of sets carry over, and others do not. In Section 2 we introduce the basic notion of a complemented set, which is used in Chapter6 to facilitate the development of the theory of measure and integration. The chapter closes with some remarks on general topology which support the conclusion that in most cases of interest it is inappropriate to define continuous functions in terms of the family of open sets.
Errett Bishop, Douglas Bridges

Chapter 4. Metric Spaces

Abstract
The concept of a metric is defined, some examples are studied, and various techniques for constructing metrics are developed. The neighborhood structure of a metric space is defined, and the notions of weakly continuous and uniformly continuous functions are introduced. Completeness is defined in Section 3, and the construction of the completion is carried through. Following Brouwer, we define a compact space to be a metric space that is complete and totally bounded. Compact and locally compact spaces are studied in Sections 4–6. Constructivizations of various classical results, such as Ascoli’s theorem, the Stone-Weierstrass theorem, and the Tietze extension theorem, are given. The concept of a located set, due to Brouwer, plays an important role. Crucial for later developments is Theorem (4.9), a partial substitute for the classical result that a closed subset of a compact space is compact.
Errett Bishop, Douglas Bridges

Chapter 5. Complex Analysis

Abstract
The constructive development of elementary complex analysis, through Cauchy’s integral formula, is a simple matter; the material seems to have a natural constructive cast. This is carried out in Sections 1, 2, 3, and 4. Next it is shown that under certain conditions it is possible to find the zeros of an analytic function. Although results of this type involve much more careful estimates than their classical counterparts, nothing basically new is required. The next section discusses singularities, and includes proofs of two constructively distinct versions of the Picard theorem on the range of a differentiable function in the neighborhood of a singularity; the proofs are based on Schottky’s theorem, the classical proof of which readily adapts to a constructive one. The last section provides a constructive version of the Riemann mapping theorem, following the classical Koebe approach as presented by Ostrowski and Warschawski. Some care must be taken in finding the right definitions, since the Riemann mapping theorem is not constructively valid without additional restrictions on the domain.
Errett Bishop, Douglas Bridges

Chapter 6. Integration

Abstract
An integration space consists of a set X with an inequality relation, a set L of partial functions from X to , and a function I: L→, called an integral, which has certain properties classically equivalent to those of a Daniell integral. Integration spaces are introduced in Section 1, and several examples are given.
Errett Bishop, Douglas Bridges

Chapter 7. Normed Linear Spaces

Abstract
In Section 1 we introduce normed linear spaces and bounded linear mappings. Section 2 is concerned with finite-dimensional spaces and with the problem of best approximation by elements of a finite-dimensional subspace. In Section 3 we discuss Lp spaces; we prove the completeness of Lp, and determine the form of the normable linear functionals on Lp in case p> 1. (In contrast to the classical theory, a bounded linear functional need not have a norm.) We then apply these results to the proof of the Radon-Nikodym theorem.
Errett Bishop, Douglas Bridges

Chapter 8. Locally Compact Abelian Groups

Abstract
Section 1 constructs Haar measure on a locally compact group G, by a method of H. Cartan. Certain least upper bounds must be proved to exist in order to make the classical proof constructive; this adds length to the classical treatment. In Section 2 convolution is defined and the group algebra is studied.
Errett Bishop, Douglas Bridges

Chapter 9. Commutative Banach Algebras

Abstract
In this chapter we introduce commutative Banach algebras and their spectra. A substitute is found for the classical result that every ideal is contained in a maximal ideal. We obtain as corollaries substitutes for other classical results, such as the compactness of the spectrum and the standard expression for the spectral norm. It is indicated in the exercises that, as far as the theory is carried, it has the same applications to analysis as its classical counterpart.
Errett Bishop, Douglas Bridges

Backmatter

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