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A geometric characterization concerning compact, convex sets

Published online by Cambridge University Press:  24 October 2008

Christopher J. Mulvey  and
Joan Wick Pelletier
Christopher J. Mulvey
Affiliation:
Mathematics Division, University of Sussex, Falmer, Brighton BN1 9QH
Joan Wick Pelletier
Affiliation:
Department of Mathematics, York University, North York, Ontario, Canada, M3J 1P3
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Extract

In this paper, we are concerned with establishing a characterization of any compact, convex set K in a normed space A in an arbitrary topos with natural number object. The characterization is geometric, not in the sense of categorical logic, but in the intuitive one, of describing any compact, convex set K in terms of simpler sets in the normed space A. It is a characterization of the compact, convex set in the sense that it provides a necessary and sufficient condition for an element of the normed space to lie within it. Having said this, we should immediately qualify our statement by stressing that this is the intuitive content of what is proved; the formal statement of the characterization is required to be in terms appropriate to the constructive context of the techniques used.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 2 , March 1991 , pp. 351 - 361
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Burden, C. W. and Mulvey, C. J.. Banach spaces in categories of sheaves. In Lecture Notes in Math. vol. 753 (Springer-Verlag, 1979), pp. 169–196.Google Scholar
[2]Johnstone, P. T.. Topos Theory (Academic Press, 1977).Google Scholar
[3]Mulvey, C. J. and Pelletier, J. Wick. A globalisation of the Hahn–Banach theorem. Adv. in. Math. (To appear.)Google Scholar
[4]Rudin, W.. Functional Analysis (McGraw-Hill, 1973).Google Scholar
[5]Vermeulen, J. J. C.. Constructive techniques in functional analysis. Ph.D. thesis, University of Sussex (1986).Google Scholar