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Finitely determined functions


We study the notion of finitely determined functions defined on a topological vector space E equipped with a biorthogonal system. We prove that, for real-valued convex functions defined on a Banach space with a Schauder basis, the notion of finitely determined function coincides with the classical continuity but outside the convex case there are many finitely determined nowhere continuous functions. This notion will be used to obtain a necessary and sufficient condition for a convex function to attain a minimum at some point. An application to the Karush–Kuhn–Tucker theorem will be given.

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  1. Albiac, F., Kalton, N.J.: Topics in Banach space theory, 2nd edn. Springer International Publishing, Switzerland (2016)

    Book  Google Scholar 

  2. Aliprentis, C.D., Border, K.C.: Infinite dimensional analysis. Springer Verlag, Berlin (1999)

    Book  Google Scholar 

  3. Bachir, M.: Limited operators and differentiability, North-Western European. J. Math. 3, 63–73 (2017)

    MATH  Google Scholar 

  4. Barbu, V., Precupanu, T.: Convexity and optimization in Banach spaces, 2nd edn. D. Reidel Publishing Co., Dordrecht (1986)

    MATH  Google Scholar 

  5. Borwein, J., Vanderwerff, J.: Convex functions: constructions, characterizations and counterexamples. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  6. Bourgain, J., Diestel, J.: Limited operators and strict cosingularity. Math. Nachr. 55–58 (1984)

  7. Carrión, H., Galindo, P., Lourenco, M.L.: Banach spaces whose bounded sets are bounding in the bidual. Ann. Acad. Sci. Fenn. Math. 31(1), 61–70 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Diestel, J.: Sequences and series in Banach spaces. Graduate texts in mathematics. Springer Verlag, Tokyo (1984)

    Book  Google Scholar 

  9. Dieudonné, J.: On biorthogonal systems. Michigan Math. J. 2(1), 7–20 (1953)

    MathSciNet  Article  Google Scholar 

  10. Enflo, P.: A counterexample to the approximation problem in Banach spaces. Acta Math. 130, 309–317 (1973)

    MathSciNet  Article  Google Scholar 

  11. Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach space theory: the basis for linear and nonlinear analysis. Springer, Canada (2011)

    Book  Google Scholar 

  12. Fabian, M., Habala, P., Hayek, P., Santalucia, V., Montesinos, P.J., Zizler, V.: Functional analysis and infinite-dimensional geometry, in CMS Books in mathematics., vol. 8. Springer, New York (2001)

    Book  Google Scholar 

  13. Kadets, M.I., Kadets, V.M.: Series in Banach spaces. Conditional and unconditional convergence, Basel-Boston-Berlin, Birkhäuser (1997)

  14. Lu, H., Freund, R.M., Nesterov, Y.: Relatively smooth convex optimization by first-order methods, and applications. SIAM J. Optim. 28(1), 333–354 (2018)

    MathSciNet  Article  Google Scholar 

  15. Marchenko, V.: Isomorphic Schauder decompositions in certain Banach spaces. Cent. Eur. J. Math. 12(11), 1714–1732 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Moreau, J.-J.: Fonctionnelles convexes, Séminaire Jean Leray (Collège de France, Paris), N. 2, 1-108 (1966–1967)

  17. Ovsepian, R., Pelczyński, A.: On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in \({L}^2\). Stud. Math. 54, 149–159 (1975)

    Article  Google Scholar 

  18. Phelps, R.R.: Convex functions, monotone operators and differentiability. Lecture notes in mathematics, vol. 1364. Springer-Verlag, Berlin (1993)

    MATH  Google Scholar 

  19. Taylor, A.B., Hendrickx, J.M., Glineur, F.: Smooth strongly convex interpolation and exact worst-case performance of first-order methods. Math. Program. 161, 307–345 (2017)

    MathSciNet  Article  Google Scholar 

  20. Yilmaz, Hasan: A generalization of multiplier rules for infinite-dimensional optimization problems. Optimization (2020).

    Article  Google Scholar 

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The third author was supported by the grants CONICYT-PFCHA/Doctorado Nacional/2018-21181905, Monge Invitation Programme of École Polytechnique, FONDECYT 1171854 and CMM Grant AFB170001.

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Correspondence to M. Bachir.

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Communicated by Enrique A. Sanchez Perez.

The authors are grateful to the anonymous referee for these valuable remarks involving the following version of the paper.

This research has been conducted within the FP2M federation (CNRS FR 2036).

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Bachir, M., Fabre, A. & Tapia-García, S. Finitely determined functions. Adv. Oper. Theory 6, 28 (2021).

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  • Schauder basis
  • Convex optimization
  • Finitely determined function
  • Directional derivatives
  • Karush–Kuhn–Tucker theorem

Mathematics Subject Classification

  • 46N10
  • 58B10
  • 49J50
  • 47A58