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Idempotent Functional Analysis: An Algebraic Approach

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This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.

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REFERENCES

  1. S. M. Avdoshin, V. V. Belov, and V. P. Maslov, Mathematical Aspects of Computational Media Synthesis [in Russian], MIEM, Moscow, 1984.

    Google Scholar 

  2. V. P. Maslov, Asymptotic Methods for Solving Pseudodi.erential Equations [in Russian], Nauka, Moscow, 1987.

    Google Scholar 

  3. V. P. Maslov, “On a new superposition principle for optimization problems,” Uspekhi Mat. Nauk [Russian Math. Surveys], 42 (1987), no. 3, 39-48.

    Google Scholar 

  4. V. P. Maslov, Méthodes opératorielles, Mir, Moscow, 1987.

    Google Scholar 

  5. S. M. Avdoshin, V. V. Belov, V. P. Maslov, and A. M. Chebotarev, “Design of computational media: mathematical aspects,” in: Mathematical Aspects of Computer Engineering (V. P. Maslov and K. A. Volosov, editors), Mir, Moscow, 1988, pp. 9-145.

    Google Scholar 

  6. Idempotent Analysis (V. P. Maslov and S. N. Samborskii, editors), Adv. Sov. Math., vol. 13, Amer. Math. Soc., Providence (R.I.), 1992.

  7. V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Application to Optimal Control [in Russian], Nauka, Moscow, 1994.

    Google Scholar 

  8. V. N. Kolokoltsov and V. P. Maslov, Idempotent Analysis and Applications, Kluwer Acad. Publ., Dordrecht, 1997.

    Google Scholar 

  9. G. L. Litvinov and V. P. Maslov, Correspondence Principle for Idempotent Calculus and Some Computer Applications (IHES/M/95/33), Institut des Hautes Études Scientifiques, Bures-sur-Yvette, 1995, in: Idempotency (J. Gunawardena, editor), Publ. of the Newton Institute, Cambridge Univ. Press, Cambridge, 1998, pp. 420-443; see also E-print http://arXiv.org/abs/math/0101021.

    Google Scholar 

  10. G. L. Litvinov and V. P. Maslov, “Idempotent mathematics: the correspondence principle and its applications to computing,” Uspekhi Mat. Nauk [Russian Math. Surveys], 51 (1996), no. 6, 209-210.

    Google Scholar 

  11. P. I. Dudnikov and S. N. Samborskii, “Endomorphisms of semimodules over semirings with idempotent operation,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 55 (1991), no. 1, 91-105.

    Google Scholar 

  12. F. L. Bacelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat, Synchronization and Linearity: an Algebra for Discrete Event Systems, J. Wiley, New York, 1992.

    Google Scholar 

  13. Idempotency (J. Gunawardena, editor), Publ. of the Newton Institute, Cambridge Univ. Press, Cambridge, 1998.

    Google Scholar 

  14. M. A. Shubin, Algebraic Remarks on Idempotent Semirings and the Kernel Theorem in Spaces of Bounded Functions [in Russian], Institute for New Technologies, Moscow, 1990; English transl. in [6].

    Google Scholar 

  15. S. C. Kleene, “Representation of events in nerve nets and.nite automata,” in: Automata Studies (J. McCarthy and C. Shannon, editors), Princeton Univ. Press, Princeton, 1956, pp. 3-40.

    Google Scholar 

  16. B. A. Carré, “An algebra for network routing problems,” J. Inst. Math. Appl., 7 (1971), 273-294.

    Google Scholar 

  17. B. A. Carré, Graphs and Networks, Oxford Univ. Press, Oxford, 1979.

    Google Scholar 

  18. M. Gondran and M. Minoux, Graphes et algorithmes, Eyrolles, Paris, 1979.

    Google Scholar 

  19. R. A. Cuningham-Green, Minimax Algebra, Springer Lect. Notes in Economics and Math. Systems, vol. 166, 1979.

  20. U. Zimmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures, vol. 10, Ann. Discrete Math., 1981.

  21. J. S. Golan, Semirings and Their Applications, Kluwer Acad. Publ., Dordrecht, 1999.

    Google Scholar 

  22. General Algebra (L. A. Skornyakov, editor) [in Russian], Mathematics Reference Library, vol. 1, Nauka, Moscow, 1990; vol. 2, ibid., 1991.

    Google Scholar 

  23. G. Birkho., Lattice Theory, Amer. Math. Soc., Providence, 1967.

  24. B. Z. Vulikh, Introduction to the Theory of Semiordered Spaces [in Russian], Fizmatgiz, Moscow, 1961.

    Google Scholar 

  25. M. M. Day, Normed Linear Spaces, Springer, Berlin, 1958.

    Google Scholar 

  26. H. H. Schaefer, Topological Vector Spaces, Macmillan, New York-London, 1966.

    Google Scholar 

  27. L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963.

    Google Scholar 

  28. P. Del Moral, “A survey of Maslov optimization theory,” in: Idempotent Analysis and Applications (V. N. Kolokoltsov and V. P. Maslov, editors), Kluwer Acad. Publ., Dordrecht, 1997.

    Google Scholar 

  29. T. Huillet, G. Rigal, and G. Salut, Optimal Versus Random Processes: a General Framework, IFAC World Congress, Tallin, USSR, 17-18 August 1990; CNRS-LAAS Report no. 242 89251, juillet 1989, 6p., Toulouse, 1989.

    Google Scholar 

  30. Proc. of the 11th Conf. on Analysis and Optimization of Systems: Discrete Event Systems (G. Gohen and J.-P. Quadrat, editors), vol. 199, Lect. Notes in Control and Inform. Sciences, Springer, 1994.

    Google Scholar 

  31. P. Del Moral, J.-Ch. Noyer, and G. Salut, “Maslov optimization theory: stochastic interpretation, particle resolution,” in: Proc. of the 11th Conf. on Analysis and Optimization of Systems: Discrete Event Systems, vol. 199, Lect. Notes in Control and Inform. Sciences, 1994, pp. 312-318.

    Google Scholar 

  32. M. Akian, J.-P. Quadrat, and M. Voit, “Bellman processes,” in: Proc. of the 11th Conf. on Analysis and Optimization of Systems: Discrete Event Systems, vol. 199, Springer Lecture Notes in Control and Inform. Sciences, 1994.

  33. J. P. Quadrat and Max-Plus working group, “Max-plus algebra and applications to system theory and optimal control,” in: Proc. of the Internat. Congress of Mathematicians, Zürich, 1994.

  34. M. Akian, J.-P. Quadrat, and M. Voit, “Duality between probability and optimization,” in: Idempotency (J. Gunawardena, editor), Publ. of the Newton Institute, Cambridge Univ. Press, Cambridge, 1998, pp. 331-353.

    Google Scholar 

  35. P. Del Moral, “Maslov optimization theory. Topological aspects,” in: Idempotency (J. Gunawardena, editor), Publ. of the Newton Institute, Cambridge Univ. Press, Cambridge, 1998, pp. 354-382.

    Google Scholar 

  36. M. Hasse, “ Über die Behandlung graphentheoretischer Probleme unter Verwendung der Matrizenrechnung,” Wiss. Z. (Techn. Univer. Dresden), 10 (1961), 1313-1316.

    Google Scholar 

  37. J. W. Leech, Classical Mechanics, Methuen, London, J. Wiley, New York, 1961.

    Google Scholar 

  38. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.

    Google Scholar 

  39. G. W. Mackey, The Mathematical Foundations of Quantum Mechanics. A Lecture-Note Volume, W. A. Benjamin, New York-Amsterdam, 1963.

    Google Scholar 

  40. E. Nelson, Probability Theory and Euclidian Field Theory, Constructive Quantum Field Theory, vol. 25, Springer, Berlin, 1973.

    Google Scholar 

  41. A. N. Vasil'ev, Functional Methods in Quantum Field Theory and Statistics [in Russian], Leningrad Gos. Univ., Leningrad, 1976.

    Google Scholar 

  42. S. Samborskii, “The Lagrange problem from the point of view of idempotent analysis,” in: Idempotency (J. Gunawardena, editor), Publ. of the Newton Institute, Cambridge Univ. Press, Cambridge, 1998, pp. 303-321.

    Google Scholar 

  43. V. P. Maslov and S. N. Samborskii, “Boundary-value problems for the stationary Hamilton-Jacobi and Bellman equations,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 49 (1997), no. 3, 433-447.

    Google Scholar 

  44. V. P. Maslov, “A new approach to generalized solutions of nonlinear systems,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 292 (1987), no. 1, 37-41.

    Google Scholar 

  45. B. Banaschewski and E. Nelson, “Tensor products and bimorphisms,” Canad. Math. Bull., 19 (1976), no. 4, 385-402.

    Google Scholar 

  46. N. Bourbaki, Topologie Générale. Éléments de Mathématique, 1ére partie, Livre III, Hermann, Paris, 1960.

    Google Scholar 

  47. V. M. Tikhomirov, “Convex analysis,” in: Current Problems in Mathematics. Fundamental Directions [in Russian], vol. 14, Itogi Nauki i Tekhniki, VINITI, Moscow, 1987, pp. 5-101.

    Google Scholar 

  48. J. R. Giles, Convex Analysis with Application in the Di.erentiation of Convex Functions, Pitman Publ., Boston-London-Melbourne, 1982.

    Google Scholar 

  49. J. Gunawardena, “An introduction to idempotency,” in: Idempotency (J. Gunawardena, editor), Publ. of the Newton Institute, Cambridge Univ. Press, Cambridge, 1998, pp. 1-49.

    Google Scholar 

  50. G. A. Sarymsakov, Topological Semifields [in Russian], Fan, Tashkent, 1969.

    Google Scholar 

  51. S. Eilenberg, Automata, Languages and Machines, Academic Press, 1974.

  52. J. H. Conway, Regular Algebra and Finite Machines, Chapman and Hall Math. Ser., London, 1971.

    Google Scholar 

  53. K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Math., Longman, 1990.

  54. A. Joyal and M. Tierney, “An extension of the Galois theory of Grothendieck,” Mem. Amer. Math. Soc., 51 (1984), no. 309, VII.

  55. S. N. Pandit, “A new matrix calculus,” SIAM J. Appl. Math., 9 (1961), no. 4, 632-639.

    Google Scholar 

  56. N. N. Vorob'ev, “The extremal matrix algebra,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 152 (1963), no. 1, 24-27.

    Google Scholar 

  57. N. N. Vorob'ev, “The extremal algebra of positive matrices” [in Russian], Elektronische Informationsverarbeitung und Kybernetik, 3 (1967), 39-71.

    Google Scholar 

  58. N. N. Vorob'ev, “The extremal algebra of nonnegative matrices” [in Russian], Elektronische Informationsverarbeitung und Kybernetik, 6 (1970), 302-312.

    Google Scholar 

  59. A. A. Korbut, “Extremal spaces,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 164 (1965), no. 6, 1229-1231.

    Google Scholar 

  60. K. Zimmermann, “A general separation theorem in extremal algebras,” Ekonomicko-Matematicky Obzor (Prague), 13 (1977), no. 2, 179-201.

    Google Scholar 

  61. G. Cohen, S. Gaubert, and J.-P. Quadrat, “Hahn-Banach separation theorem for Max-Plus semimodules,” to appear in: Proc. of “Conférence en l'honneur du Professeur Alain Bensoussan à l'occasion de son 60ème anniversaire, 4-5 dec. 2000, Paris, to be published by IOS, the Netherlands.

  62. P. Butkovič, “Strong regularity of matrices-A survey of results,” Discrete Applied Math., 48 (1994), 45-68.

    Google Scholar 

  63. V. Kolokoltsov, “Idempotent structures in optimization,” in: Surveys in Modern Mathematics and Applications, vol. 65, Proc. of L. S. Pontryagin Conference, Optimal Control, Itogi Nauki i Tehniki, VINITI, Moscow, 1999, pp. 118-174.

    Google Scholar 

  64. R. Bellman and W. Karush, “On a new functional transform in analysis: the maximum transform,” Bull. Amer. Math. Soc., 67 (1961), 501-503.

    Google Scholar 

  65. I. V. Romanovskii, “Optimization of the stationary control for a discrete deterministic process” [in Russian], Kibernetika (1967), no. 2, 66-78.

    Google Scholar 

  66. I. V. Romanovskii, “Asymptotic behavior of a discrete deterministic process with a continuous state space” [in Russian], Optimal planning, Trudy IM SO Akademii Nauk SSSR, 8 (1967), 171-193.

    Google Scholar 

  67. E. Schrödinger, “Quantization as an eigenvalue problem” [in German], Annalen der Physik, 364 (1926), 361-376.

    Google Scholar 

  68. E. Hopf, “The partial differential equation u t + uu x = µu xx ,” Comm. Pure Appl. Math., 3 (1950), 201-230.

    Google Scholar 

  69. P.-L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, 1982.

    Google Scholar 

  70. M. G. Crandall, H. Ishii, and P.-L. Lions, “A user's guide to viscosity solutions,” Bull. Amer. Math. Soc., 27 (1992), 1-67.

    Google Scholar 

  71. W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993.

    Google Scholar 

  72. A. I. Subbotin, “Minimax solutions of first-order partial differential equations,” Uspekhi Mat. Nauk [Russian Math. Surveys], 51 (1996), no. 2, 105-138.

    Google Scholar 

  73. Viscosity Solutions and Applications (I. Capuzzo Dolcetta and P.-L. Lions, editors), vol. 1660, Lecture Notes in Math., Springer, 1997.

  74. P. D. Lax, “Hyperbolic systems of conservation laws II,” Comm. Pure Appl. Math., 10 (1957), 537-566.

    Google Scholar 

  75. O. A. Oleinik, “Discontinuous solutions of nonlinear differential equations” [in Russian], Uspekhi Mat. Nauk, 12 (1957), no. 3, 3-73.

    Google Scholar 

  76. O. Viro, “Dequantization of real algebraic geometry on a logarithmic paper,” 3rd European Congress of Mathematics, E-print http://arXiv.org/abs/math/0005163.

  77. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Linear functionals on idempotent spaces. Algebraic approach,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.]. 363 (1998), no. 3, 298-300; see also E-print http://arXiv.org/abs/math/0012268.

    Google Scholar 

  78. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, Idempotent Functional Analysis: An Algebraic Approach [in Russian], Internat. Sophus Lie Center, Moscow, 1998.

    Google Scholar 

  79. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, Idempotent Functional Analysis: An Algebraic Approach, E-print http://arXiv.org/abs/math/0009128. 2000.

  80. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Tensor products of idempotent semimodules. An algebraic approach,” Mat. Zametki [Math. Notes], 65 (1999), no. 4, 572-585; see also E-print http: // arXiv.org/abs/math/0101153.

    Google Scholar 

  81. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, vol. 16, Mem. Amer. Math. Soc, Amer. Math. Soc., Providence, 1955.

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  1. International Sophus Lie Center, Russia

    G. L. Litvinov & G. B. Shpiz

  2. M. V. Lomonosov Moscow State University, Russia

    V. P. Maslov

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Litvinov, G.L., Maslov, V.P. & Shpiz, G.B. Idempotent Functional Analysis: An Algebraic Approach. Mathematical Notes 69, 696–729 (2001). https://doi.org/10.1023/A:1010266012029

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