Abstract

The classical Weierstrass theorem states that every continuous function ff defined on a compact set Ω⊂Rn\Omega \subset\mathbb{R}^n can be uniformly approximated by polynomials. We show first that it is again valid if Ω\Omega is a compact Hausdorff metric space, i.e., it holds in the following sense: there exists a surjective isometry TT from a compact set KΩK_\Omega of a Banach sequence space SS to Ω\Omega, such that for every ε>0\varepsilon>0 there is an nn variable polynomial pp satisfying ∣f(T(s))−p(s1,s2,⋯ ,sn)∣<ε,  ∀s=(sj)∈KΩ.|f(T(s))-p(s_1,s_2,\cdots,s_n)|<\varepsilon,\;\forall s=(s_j)\in K_{\Omega}. We prove also that for any weakweak (w∗w^*, resp.) continuous positively homogenous function ff defined on a (dual, resp.) Banach space XX (X∗X^*, resp.) then for all ε>0\varepsilon>0 and for every weakly compact set K⊂XK\subset X( w∗w^* compact set K⊂X∗K\subset X^*), there exist ϕi∈X∗\phi_i\in X^* (X,X, resp.) for i=1,2,⋯ ,m,i=1,2,\cdots, m, and ψj∈X∗\psi_j\in X^* (X,X, resp.) for j=1,2,⋯ ,nj=1,2, \cdots,n such that ∣f(x)−[(ϕ1∨ϕ2∨⋯∨ϕm)(x)−(ψ1∨ψ2∨⋯∨ψn)(x)]∣<ε|f(x)-[(\phi_1\vee\phi_2\vee\cdots\vee\phi_m)(x)- (\psi_1\vee\psi_2\vee\cdots\vee\psi_n)(x)]|<\varepsilon uniformly for x∈K.x\in K. Let cc(X)cc(X) (wcc(X)wcc(X), reps.) be the norm semigroup consisting of all nonempty (weakly, resp.) compact convex sets of the space XX. As its application, we give two representation theorems of the duals of cc(X)cc(X) and wcc(X)wcc(X).

Author information

Contact details are reproduced from the original publication and may be historical.

Lixin Cheng

School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China

lxcheng@xmu.edu.cn

Yu Zhou

School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China

roczhoufly@126.com

Keywords

  • Weierstrass theorem
  • function approximation
  • weakly continuous function
  • weakly compact set
  • normed semigroup
  • Delta-convex polyhedron support function

Mathematics Subject Classification

41A10, 41A30, 41A65, 46A20; 46B20, 46E05, 46J10

References

  1. S. Banach: A Course of Functional Analysis, Radyans’ka Shkola, Kiev (1948) (in Ukrainian).Google Scholar
  2. Y. Benyamini, J. Lindenstauss: Geometric Nonlinear Functional Analysis, Colloq. Publ. 48, Amer. Math. Soc., Providence (2000).https://doi.org/10.1090/coll/048Google Scholar
  3. M. C. Boiso: Approximation of Lipschitz functions by ∆-convex functions in Banach spaces, Israel J. Math. (1998) 269–284.https://doi.org/10.1007/BF02773472Google Scholar
  4. L. Cheng, Q. Cheng, Z. Luo: On some new characterizations of weakly compact sets in Banach spaces, Stud. Math. 201 (2010) 155–166.https://doi.org/10.4064/sm201-2-3Google Scholar
  5. L. Cheng, Q. Cheng, Z. Luo, W. Zhang: Every weakly compact set can be uniformly embedded into a reflexive Banach space, Acta Math. Sin., Engl. Ser. 25(7) (2009) 1109–1112.https://doi.org/10.1007/s10114-009-7545-5Google Scholar
  6. L. Cheng, Q. Cheng, B. Wang, W. Zhang: On super weakly compact sets and uniformly convexifiable sets, Stud. Math. 199(2) (2010) 145–169.https://doi.org/10.4064/sm199-2-2Google Scholar
  7. L. Cheng, C. Wu, X. Xue, X. Yao: Convex functions, subdifferentiability and renormings, Acta Math. Sin., New Ser. 14(1) (1998) 47–56.https://doi.org/10.1007/BF02563883Google Scholar
  8. B. De, S. Francesco, T. Hu, J. Huang: Weak∗ -topology and Alaoglu’s theorem on hyperspace, J. Nonlinear Convex Anal. 10(1) (2009) 33–40.Google Scholar
  9. R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific & Technical, Harlow (1993).Google Scholar
  10. B. Fuchssteiner, W. Lusky: Convex Cones, North Holland, Amsterdam (1981).Google Scholar
  11. T. Hu, J. Huang: Weak and weak* topologies and Brodskii-Milman’s theorem on hyperspaces, Taiwanese J. Math. 13(2A) (2009) 459–466.https://doi.org/10.11650/twjm/1500405349Google Scholar
  12. T. Hu, J. Huang: Weak and strong convergence in the hyperspace CC(X), Taiwanese J. Math. 12(5) (2008) 1285–1291.https://doi.org/10.11650/twjm/1500574263Google Scholar
  13. T. Hu, J. Fang: Weak topology and Browder-Kirk’s theorem on hyperspace, J. Math. Anal. Appl. 334(2) (2007) 799–803.https://doi.org/10.1016/j.jmaa.2006.12.078Google Scholar
  14. K. John, V. Zizler: A renorming of dual spaces, Israel J. Math. 12 (1972) 331–336.https://doi.org/10.1007/BF02764623Google Scholar
  15. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces. I: Sequence Spaces, Springer, Berlin (1977).https://doi.org/10.1007/978-3-642-66557-8Google Scholar
  16. S. Kakutani, Weak topology, bicompact set and the principle of duality, Proc. Imp. Acad. Jap. 16 (1940) 63–67.https://doi.org/10.3792/pia/1195579206Google Scholar
  17. K. Keimel, W. Roth: Ordered Cones and Approximation, Springer, Berlin (1992).https://doi.org/10.1007/BFb0089190Google Scholar
  18. K. Keimel, W. Roth: A Korovkin type approximation theorem for set-valued functions, Proc. Amer. Math. Soc. 104 (1988) 819–824.https://doi.org/10.1090/S0002-9939-1988-0964863-8Google Scholar
  19. M. Krein, S. Krein: On an inner characteristic of the set of all continuous functions defined on a bicompact Hausdorff space, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 27 (1940) 427–430.Google Scholar
  20. H. Radstrom: An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169.https://doi.org/10.1090/S0002-9939-1952-0045938-2Google Scholar
  21. W. Roth: Real and complex linear extensions for locally convex cones, J. Funct. Anal. 151(2) (1997) 437–454.https://doi.org/10.1006/jfan.1997.3142Google Scholar
  22. M. Stone: Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41(3) (1937) 375–481.https://doi.org/10.1090/S0002-9947-1937-1501905-7Google Scholar
  23. M. Stone: The generalized Weierstrass approximation theorem, Math. Mag. 21(4) (1948) 167–184; 21(5) (1948) 237–254.https://doi.org/10.2307/3029337Google Scholar
  24. S. Troyanski: On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Stud. Math. 37 (1971) 173–180.https://doi.org/10.4064/sm-37-2-173-180Google Scholar
  25. K. Weierstrass: Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Berl. Ber. (1885) 633–640; 789–806.Google Scholar
  26. K. Yosida: Functional Analysis, 6th Ed., Springer, Berlin (1999).Google Scholar

Suggested citation

L. Cheng, Y. Zhou. “On Approximation by Δ-Convex Polyhedron Support Functions and the Dual of cc(X) and wcc(X).” Journal of Convex Analysis 19 (2012), No. 1, 201–212. https://doi.org/10.68381/jca19012

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.