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A Representation Theorem for Riesz Spaces and Its Applications to Economics

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  • Abramovich, Y A
  • Aliprantis, C D
  • Zame, W R

Abstract

We show that a Dedekind complete Riesz space which contains a weak unit e and admits a strictly positive order continuous linear functional can be represented as a subspace of the space L(subscript "1") of integrable functions on a probability measure space in such a way that the order ideal generated by e is carried onto L(subscript "infinity"). As a consequence, we obtain a characterization of abstract M-spaces that are isomorphic to concrete L(subscript "infinity")-spaces. Although these results are implicit in the literature on representation of Riesz spaces, they are not available in this form. This research is motivated by, and has applications in, general equilibrium theory in infinite dimensional spaces.

Suggested Citation

  • Abramovich, Y A & Aliprantis, C D & Zame, W R, 1995. "A Representation Theorem for Riesz Spaces and Its Applications to Economics," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(3), pages 527-535, May.
  • Handle: RePEc:spr:joecth:v:5:y:1995:i:3:p:527-35
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    1. Peleg, Bezalel & Yaari, Menahem E, 1970. "Markets with Countably Many Commodities," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 11(3), pages 369-377, October.
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    6. Aliprantis, Charalambos D & Brown, Donald J & Burkinshaw, Owen, 1987. "Edgeworth Equilibria," Econometrica, Econometric Society, vol. 55(5), pages 1109-1137, September.
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    Cited by:

    1. Simone Cerreia Vioglio & Fabio Maccheroni & Massimo Marinacci, 2015. "Hilbert A-Modules," Working Papers 544, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    2. Simone Cerreia Vioglio & Fabio Maccheroni & Massimo Marinacci, 2016. "Orthogonal Decompositions in Hilbert A-Modules," Working Papers 577, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.

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