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Constructive analysis. (English) Zbl 0656.03042

Grundlehren der Mathematischen Wissenschaften, 279. Berlin etc.: Springer-Verlag. XII, 477 p.; DM 138.00 (1985).
In a most radical departure from the mainstream of twentieth century research, E. Bishop’s “Foundations of constructive analysis (1967; Zbl 0183.015) demonstrated the possibility of “a straightforward realistic approach to mathematics”. Although the number of workers to follow this approach during the subsequent twenty years was far less than might have been expected, advances by a few active constructivists were extensive enough to render obsolete certain portions of Bishop’s inestimable treatise. Douglas Bridges has contributed many of these advances, and has revised and extended Bishop’s book. The most important changes and additions concern the following topics: integration theory, Banach algebras, the Riemann mapping theorem. Bishop’s work on martingales and ergodic theory has not been included, pending revision in a separate paper.
As Bridges states, this book is not a substitute for a course in classical abstract analysis; experience in the classical treatment is necessary to attain the level of maturity required for an appreciation of the constructive approach. Furthermore, the classical theory forms, at least for the present time, the basis for a full constructive development. As Bishop states in his original preface: “We are not contending that idealistic mathematics is worthless from the constructive point of view. This would be as silly as contending that unrigorous mathematics is worthless from the classical point of view. Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof.”
Reviewer: M.Mandelkern

MSC:

03F60 Constructive and recursive analysis
03F65 Other constructive mathematics
46S30 Constructive functional analysis
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures

Citations:

Zbl 0183.015