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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 596384, 8 pages
http://dx.doi.org/10.1155/2014/596384
Research Article

A Characterization of Completeness via Absolutely Convergent Series and the Weierstrass Test in Asymmetric Normed Semilinear Spaces

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia
2Departamento de Ciencias Matemáticas e Informática, Universidad de las Islas Baleares, Carretera de Valldemossa km. 7.5, 07122 Palma de Mallorca, Spain

Received 4 April 2014; Revised 9 June 2014; Accepted 13 June 2014; Published 10 July 2014

Academic Editor: J. J. Font

Copyright © 2014 N. Shahzad and O. Valero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. J. Duffin and L. A. Karlovitz, “Formulation of linear programs in analysis. I. Approximation theory,” SIAM Journal on Applied Mathematics, vol. 16, pp. 662–675, 1968. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. G. Krein and A. A. Nudelman, The Markov Moment Problem and Extremal Problems, American Mathematical Society, 1977. View at MathSciNet
  3. S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Birkhauser, Basel, Switzerland, 2013. View at MathSciNet
  4. C. Alegre, “Continuous operators on asymmetric normed spaces,” Acta Mathematica Hungarica, vol. 122, no. 4, pp. 357–372, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. C. Alegre and I. Ferrando, “Quotient subspaces of asymmetric normed linear spaces,” Boletín de la Sociedad Matemática Mexicana, vol. 13, no. 2, pp. 357–365, 2007. View at MathSciNet · View at Scopus
  6. C. Alegre, I. Ferrando, L. M. García-Raffi, and E. A. Sánchez Pérez, “Compactness in asymmetric normed spaces,” Topology and its Applications, vol. 155, no. 6, pp. 527–539, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. C. Alegre, J. Ferrer, and V. Gregori, “Quasi-uniform structures in linear lattices,” The Rocky Mountain Journal of Mathematics, vol. 23, no. 3, pp. 877–884, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. C. Alegre, J. Ferrer, and V. Gregori, “On the Hahn-Banach theorem in certain linear quasi-uniform structures,” Acta Mathematica Hungarica, vol. 82, no. 4, pp. 325–330, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. J. Ferrer, V. Gregori, and C. Alegre, “Quasi-uniform structures in linear lattices,” The Rocky Mountain Journal of Mathematics, vol. 23, no. 3, pp. 877–884, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. L. M. García-Raffi, “Compactness and finite dimension in asymmetric normed linear spaces,” Topology and its Applications, vol. 153, no. 5-6, pp. 844–853, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. L. M. García-Raffi and S. Romaguera, “Sequence spaces and asymmetric norms in the theory of computational complexity,” Mathematical and Computer Modelling, vol. 36, no. 1-2, pp. 1–11, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. L. M. García-Raffi, S. Romaguera, and E. A. Sánchez-Pérez, “The bicompletion of an asymmetric normed linear space,” Acta Mathematica Hungarica, vol. 97, no. 3, pp. 183–191, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. L. M. García-Raffi and R. Sanchez-Pérez, “The dual space of an asymmetric normed linear space,” Quaestiones Mathematicae. Journal of the South African Mathematical Society, vol. 26, no. 1, pp. 83–96, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. L. M. García-Raffi, S. Romaguera, and E. A. Sánchez-Pérez, “The supremum asymmetric norm on sequence spaces: a general framework to measure complexity distances,” Electronic Notes in Theoretical Computer Science, vol. 74, pp. 39–50, 2003. View at Publisher · View at Google Scholar
  15. L. M. García Raffi, S. Romaguera, and E. A. Sánchez Pérez, “Weak topologies on asymmetric normed linear spaces and non-asymptotic criteria in the theory of complexity analysis of algorithms,” Journal of Analysis and Applications, vol. 2, no. 3, pp. 125–138, 2004. View at MathSciNet
  16. J. Rodríguez-López and S. Romaguera, “Closedness of bounded convex sets of asymmetric normed linear spaces and the Hausdorff quasi-metric,” Bulletin of the Belgian Mathematical Society: Simon Stevin, vol. 13, no. 3, pp. 551–562, 2006. View at MathSciNet · View at Scopus
  17. S. Romaguera, E. A. Sánchez-Pérez, and O. Valero, “A characterization of generalized monotone normed cones,” Acta Mathematica Sinica, vol. 23, no. 6, pp. 1067–1074, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. S. Romaguera, E. A. Sánchez Pérez, and O. Valero, “Dominated extensions of functionals and V-convex functions on cancellative cones,” Bulletin of the Australian Mathematical Society, vol. 67, no. 1, pp. 87–94, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. S. Romaguera and M. Sanchis, “Semi-Lipschitz functions and best approximation in quasi-metric spaces,” Journal of Approximation Theory, vol. 103, no. 2, pp. 292–301, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. S. Romaguera and M. Schellekens, “Duality and quasi-normability for complexity spaces,” Applied General Topology, vol. 3, no. 1, pp. 91–112, 2002. View at MathSciNet
  21. H. P. A. Künzi, “Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology,” in Handbook of the History of General Topology, C. E. Aull and R. Lowen, Eds., vol. 3, pp. 853–968, Kluwer Academic, New York, NY, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  22. E. A. Ok, Real Analysis with Economic Applications, Princeton University Press, 2007. View at MathSciNet
  23. G. J. O. Jameson, Topology and Normed Spaces, Chapman and Hall, London, UK, 1974. View at MathSciNet
  24. J. J. Conradie and M. D. Mabula, “Convergence and left-K-sequential completeness in asymmetrically normed lattices,” Acta Mathematica Hungarica, vol. 139, no. 1-2, pp. 147–159, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. I. L. Reilly, P. V. Subrahmanyam, and M. K. Vamanamurthy, “Cauchy sequences in quasi-pseudo-metric spaces,” Monatshefte für Mathematik, vol. 93, no. 2, pp. 127–140, 1982. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. M. D. Mabula, Compactness in asymmetric normed lattices [Ph.D. thesis], Department of Mathematics and Applied Mathematics, University of Cape Town, 2012.
  27. D. H. Griffel, Applied Functional Analysis, Dover, 1985. View at MathSciNet
  28. J. L. Kelley, I. Namioka, and W. F. Donoghue Jr., Linear Topological Spaces, D. Van Nostrand, 1963. View at MathSciNet