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Non-standard analysis. (English) Zbl 0151.00803
Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Company. xi, 293 p. (1966).
The author delights in showing that the old non-rigorous approach to analysis by means of infinitesimals can, in many respects, be justified and expanded by use of non-standard models for analysis. The latter are models which satisfy the same elementary properties (i.e. those expressible in the first-order predicate calculus) as the real number system but are not isomorphic to that system. Topics covered include: tools from logic, differential and integral calculus, general topology, functions of real and complex variables, linear spaces, topological groups and Lie groups, variational problems, hydrodynamics, and the history of the calculus. A striking application is the solution by A. R. Bernstein and the author [Pac. J. Math. 16, 421–431 (1966; Zbl 0141.12903)] of an open invariant subspace problem of P. R. Halmos and K. T. Smith.
Reviewer: Elliott Mendelson

MSC:
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03Hxx Nonstandard models
26E35 Nonstandard analysis
26-02 Research exposition (monographs, survey articles) pertaining to real functions
26-03 History of real functions
01A65 Development of contemporary mathematics
28E05 Nonstandard measure theory
30G06 Non-Archimedean function theory
46S20 Nonstandard functional analysis
47S20 Nonstandard operator theory
54J05 Nonstandard topology