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On the Nonstandard Duality Theory of Locally Convex Spaces

Published online by Cambridge University Press:  20 November 2018

Arthur D. Grainger
Arthur D. Grainger*
Affiliation:
Louisiana State UniversityBaton Rouge, Louisiana
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This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).

The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 460 - 479
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Comfort, W. W. and Negrepontis, S., The theory of ultrafilters (Springer-Verlag, New York, 1974).Google Scholar
2. Grainger, A. D., Finite points of filters in infinite dimensional vector spaces, Fundamenta Mathematica. 104 (1979), 47–67.Google Scholar
3. Henson, C. W. and Moore, L. C., Jr., The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 405–435.Google Scholar
4. Henson, C. W. and Moore, L. C., Jr., Invariance of the nonstandard hulls of locally convex spaces, Duke Math. J.. 40 (1973), 193–206.Google Scholar
5. Horvath, J., Topological vector spaces and distributions vol. 1 (Addison Wesley, Reading, Mass., 1966).Google Scholar
6. Luxemburg, W. A. J., A general theory of monads, W. A. J. Luxemburg, ed., in Applications of model theory to algebra, analysis and probability (Holt, Rinehart and Winston, New York, 1969), 18–86.Google Scholar
7. Robinson, A., Non-standard analysis (North-Holland, Amsterdam, 1966).Google Scholar