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The range of an o-weakly compact mapping

  • P. G. Dodds (a1)

Abstract

It is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

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References

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[1]Aliprantis, C. D. and Burkinshaw, O., Locally solid Riesz spaces, (Academic Press, New York, San Francisco, London, 1978).
[2]Choquet, G., Lectures on analysis, Eds. Marsden, , Lance, and Gelbart, , (W. A. Benjamin, New York, Amsterdam, 1969).
[3]Diestel, J. and Uhl, J. J. Jr, ‘Vector measures’, Math. Surueys 15 (Amer. Math. Soc., 1977).
[4]Dodds, P. G., ‘o-weakly compact mappings of Riesz spaces’, Trans. Amer. Math. Soc. 214 (1975), 389402.
[5]Kluv´nek, I., ‘The range of a vector-valued measure’, Math. Systems Theory 7 (1973), 4454.
[6]Kluv´nek, I., ‘Characterization of the closed convex hull of the range of a vector-valued measure’, J. Funct. Anal. 21 (1976), 316329.
[7]Kluv´nek, I., ‘Conical measures and vector measures’, Ann. Inst. Fourier (Grenoble) 27 1 (1977), 83105.
[8]Kluv´nek, I. and Knowles, G., Vector measures and control systems (North-Holland Publishing Co., Amsterdam, 1975).
[9]Luxemburg, W. A. J., Some aspects of the theory of Riesz spaces (The University of Arkansas Lecture Notes in Mathematics, # 4, Fayetteville, 1979).
[10]Luxemberg, W. A. J. and Zaanen, A. C., ‘Notes on Banach function spaces VI-XIII’, Nederl. Akad. Wetensch. Proc. Ser. A 66 (1963), 251263, 496–504, 655–681; 67 (1964), 104–119, 360–376, 493–518, 519–543.
[11]Luxemberg, W. A. J. and Zaanen, A. C., Riesz spaces (North-Holland, 1971).
[12]Segal, I. E., ‘Equivalences of measure spaces’, Amer. J. Math. 73 (1951), 275313.
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The range of an o-weakly compact mapping

  • P. G. Dodds (a1)

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