Nonstandard Analysis and Vector Lattices

1. Front Matter

   Pages i-xii

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2. Nonstandard Methods and Kantorovich Spaces

   • A. G. Kusraev, S. S. Kutateladze

   Pages 1-79

3. Functional Representation of a Boolean Valued Universe

   • A. E. Gutman, G. A. Losenkov

   Pages 81-104

4. Dual Banach Bundles

   • A. E. Gutman, A. V. Koptev

   Pages 105-159

5. Infinitesimals in Vector Lattices

   • È. Yu. Emel’yanov

   Pages 161-230

6. Vector Measures and Dominated Mappings

   • A. G. Kusraev, S. A. Malyugin

   Pages 231-299

7. Back Matter

   Pages 300-307

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Nonstandard methods of analysis consist generally in comparative study of two interpretations of a mathematical claim or construction given as a formal symbolic expression by means of two different set-theoretic models: one, a "standard" model and the other, a "nonstandard" model. The second half of the twentieth century is a period of significant progress in these methods and their rapid development in a few directions. The first of the latter appears often under the name coined by its inventor, A. Robinson. This memorable but slightly presumptuous and defiant term, non­ standard analysis, often swaps places with the term Robinsonian or classical non­ standard analysis. The characteristic feature of Robinsonian analysis is a frequent usage of many controversial concepts appealing to the actual infinitely small and infinitely large quantities that have resided happily in natural sciences from ancient times but were strictly forbidden in modern mathematics for many decades. The present-day achievements revive the forgotten term infinitesimal analysis which reminds us expressively of the heroic bygones of Calculus. Infinitesimal analysis expands rapidly, bringing about radical reconsideration of the general conceptual system of mathematics. The principal reasons for this progress are twofold. Firstly, infinitesimal analysis provides us with a novel under­ standing for the method of indivisibles rooted deeply in the mathematical classics.

• functional analysis
• mathematical logic
• set theory

• Siberian Division of the Russian Academy of Sciences, Sobolev Institute of Mathematics, Novosibirsk, Russia

  S. S. Kutateladze

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