AN ANALOGY OF HAHN–BANACH SEPARATION THEOREM FOR NEARLY TOPOLOGICAL LINEAR SPACES

Madhu Ram     (Department of Mathematics, University of Jammu, Jammu-180006, India)

Abstract


In this paper, we introduce the notion of nearly topological linear spaces and use it to formulate an alternative definition of the Hahn–Banach separation theorem. We also give an example of a topological linear space to which the result is not valid. It is shown that \(\mathbb{R}\) with its ordinary topology is not a nearly topological linear space.

Keywords


Hahn–Banach separation theorem, \(\alpha\)-open sets, \(\alpha\)-compact sets, Nearly topological linear spaces

Full Text:

PDF

References


  1. Amar A.B., Cherif M.A., Mnif M. Fixed-point theory on a Frechet topological vector space. Int. J. Math. Math. Sci., 2011. Vol. 2011. Art. no. 390720. P. 1–9. DOI: 10.1155/2011/390720
  2. Deutsch F.R., Maserick P.H. Applications of the Hahn–Banach theorem in approximation theory. SIAM Rev., 1967. Vol. 9, No. 3. P. 516–530. URL: https://www.jstor.org/stable/2027994
  3. Deutsch F., Hundal H., Zikatanov L. Some Applications of the Hahn–Banach Separation Theorem, 2017. 26 p. arXiv:1712.10250v1 [math.FA]
  4. Helton J.W., Klep I., McCullough S. The Tracial Hahn–Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra, 2014. 56 p. arXiv:1407.8198 [math.OA]
  5. Luna-Elizarrarás M.E., Perez-Regalado C.O., Shapiro M. On linear functionals and Hahn-Banach theorems for hyperbolic and bicomplex modules. Adv. Appl. Clifford Algebr., 2014. Vol. 24. P. 1105–1129. DOI: 10.1007/s00006-014-0503-z
  6. Maheshwari S.N., Thakur S.S. On α-compact spaces. Bull. Inst. Math. Acad. Sin. (N.S.), 1985. Vol. 13, No. 4. P. 341–347.
  7. Njåstad O. On some classes of nearly open sets. Pacific J. Math., 1965. Vol. 15, No. 3. P. 961–970.
  8. Nörtemann S. The Hahn-Banach theorem for partially ordered totally convex, positively convex and superconvex modules. Appl. Categ. Structures, 2002. Vol. 10, p. 417–429. DOI: 10.1023/A:1016390813177
  9. Rudin W. Functional Analysis. 2nd ed. Singapore: McGraw-Hill Inc., 1991. 448 p.
  10. Schaefer H.H. Topological Vector Spaces. New York: Springer-Verlag, 1971. 296 p. DOI: 10.1007/978-1-4684-9928-5




DOI: http://dx.doi.org/10.15826/umj.2021.1.006

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.