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Nonstandard extensions of uniform algebraic systems

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Abstract

Important applications of nonstandard analysis to the theory of Banach spaces are based on the construction of the nonstandard hull of a normed linear space [1,2]. By making use of the iterated nonstandard enlargements [3], in the present article we propose a universal construction for arbitrary uniform algebraic systems which allows one to study the nonstandard hulls and completions of such systems.

In Section 1 we give necessary information about nonstandard enlargements and prove a number of important results in the general theory of monads. In Section 2 we present basic facts on nonstandard topologies and uniform algebraic systems. In Section 3 we describe a general algebraic construction with the help of which we study the question of completing uniform algebraic systems. Conditions for existence of nonstandard hulls for such systems are obtained in Section 4.

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Authors and Affiliations

  1. Saratov

    V. A. Molchanov

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  1. V. A. Molchanov
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Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1094–1105, September–October, 1994.

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Molchanov, V.A. Nonstandard extensions of uniform algebraic systems. Sib Math J 35, 976–985 (1994). https://doi.org/10.1007/BF02104575

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  • DOI: https://doi.org/10.1007/BF02104575

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